code-stellarator-optimization/include/sctl/boundary_quadrature.hpp

#ifndef _SCTL_BOUNDARY_QUADRATURE_HPP_
#define _SCTL_BOUNDARY_QUADRATURE_HPP_

#include <mutex>
#include <atomic>
#include <tuple>

namespace SCTL_NAMESPACE {

template <class Real, Integer DIM, Integer ORDER> class Basis {
  public:
    using ValueType = Real;

    // class EvalOperator {
    //   public:
    // };
    using EvalOpType = Matrix<ValueType>;

    static constexpr Long Dim() {
      return DIM;
    }
    static constexpr Long Size() {
      return pow<DIM,Long>(ORDER);
    }
    static const Matrix<ValueType>& Nodes() {
      static Matrix<ValueType> nodes_(DIM,Size());
      auto nodes_1d = [](Integer i) {
        return 0.5 - 0.5 * sctl::cos<ValueType>((2*i+1) * const_pi<ValueType>() / (2*ORDER));
      };
      { // Set nodes_
        static std::mutex mutex;
        static std::atomic<Integer> first_time(true);
        if (first_time.load(std::memory_order_relaxed)) {
          std::lock_guard<std::mutex> guard(mutex);
          if (first_time.load(std::memory_order_relaxed)) {
            Integer N = 1;
            for (Integer d = 0; d < DIM; d++) {
              for (Integer j = 0; j < ORDER; j++) {
                for (Integer i = 0; i < N; i++) {
                  for (Integer k = 0; k < d; k++) {
                    nodes_[k][j*N+i] = nodes_[k][i];
                  }
                  nodes_[d][j*N+i] = nodes_1d(j);
                }
              }
              N *= ORDER;
            }

            std::atomic_thread_fence(std::memory_order_seq_cst);
            first_time.store(false);
          }
        }
      }
      return nodes_;
    }
    static const Vector<ValueType>& QuadWts() {
      static Vector<ValueType> wts(Size());
      { // Set nodes_
        static std::mutex mutex;
        static std::atomic<Integer> first_time(true);
        if (first_time.load(std::memory_order_relaxed)) {
          std::lock_guard<std::mutex> guard(mutex);
          if (first_time.load(std::memory_order_relaxed)) {
            StaticArray<ValueType,ORDER> wts_1d;
            { // Set wts_1d
              Vector<ValueType> x_(ORDER);
              ChebBasis<ValueType>::template Nodes<1>(ORDER, x_);

              Vector<ValueType> V_cheb(ORDER * ORDER);
              { // Set V_cheb
                Vector<ValueType> I(ORDER*ORDER);
                I = 0;
                for (Long i = 0; i < ORDER; i++) I[i*ORDER+i] = 1;
                ChebBasis<ValueType>::template Approx<1>(ORDER, I, V_cheb);
              }
              Matrix<ValueType> M(ORDER, ORDER, V_cheb.begin());

              Vector<ValueType> w_sample(ORDER);
              for (Integer i = 0; i < ORDER; i++) {
                w_sample[i] = (i % 2 ? 0 : -(ORDER/(ValueType)(i*i-1)));
              }
              for (Integer j = 0; j < ORDER; j++) {
                wts_1d[j] = 0;
                for (Integer i = 0; i < ORDER; i++) {
                  wts_1d[j] += M[j][i] * w_sample[i] / ORDER;
                }
              }
            }

            wts[0] = 1;
            Integer N = 1;
            for (Integer d = 0; d < DIM; d++) {
              for (Integer j = 1; j < ORDER; j++) {
                for (Integer i = 0; i < N; i++) {
                  wts[j*N+i] = wts[i] * wts_1d[j];
                }
              }
              for (Integer i = 0; i < N; i++) {
                wts[i] *= wts_1d[0];
              }
              N *= ORDER;
            }

            std::atomic_thread_fence(std::memory_order_seq_cst);
            first_time.store(false);
          }
        }
      }
      return wts;
    }

    static void Grad(Vector<Basis>& dX, const Vector<Basis>& X) {
      static Matrix<ValueType> GradOp[DIM];
      static std::mutex mutex;
      static std::atomic<Integer> first_time(true);
      if (first_time.load(std::memory_order_relaxed)) {
        std::lock_guard<std::mutex> guard(mutex);
        if (first_time.load(std::memory_order_relaxed)) {
          { // Set GradOp
            auto nodes = Basis<ValueType,1,ORDER>::Nodes();
            SCTL_ASSERT(nodes.Dim(1) == ORDER);
            Matrix<ValueType> M(ORDER, ORDER);
            for (Integer i = 0; i < ORDER; i++) { // Set M
              Real x = nodes[0][i];
              for (Integer j = 0; j < ORDER; j++) {
                M[j][i] = 0;
                for (Integer l = 0; l < ORDER; l++) {
                  if (l != j) {
                    Real M_ = 1;
                    for (Integer k = 0; k < ORDER; k++) {
                      if (k != j && k != l) M_ *= (x - nodes[0][k]);
                      if (k != j) M_ /= (nodes[0][j] - nodes[0][k]);
                    }
                    M[j][i] += M_;
                  }
                }
              }
            }
            for (Integer d = 0; d < DIM; d++) {
              GradOp[d].ReInit(Size(), Size());
              GradOp[d] = 0;
              Integer stride0 = sctl::pow<Integer>(ORDER, d);
              Integer repeat0 = sctl::pow<Integer>(ORDER, d);
              Integer stride1 = sctl::pow<Integer>(ORDER, d+1);
              Integer repeat1 = sctl::pow<Integer>(ORDER, DIM-d-1);
              for (Integer k1 = 0; k1 < repeat1; k1++) {
                for (Integer i = 0; i < ORDER; i++) {
                  for (Integer j = 0; j < ORDER; j++) {
                    for (Integer k0 = 0; k0 < repeat0; k0++) {
                      GradOp[d][k1*stride1 + i*stride0 + k0][k1*stride1 + j*stride0 + k0] = M[i][j];
                    }
                  }
                }
              }
            }
          }
          std::atomic_thread_fence(std::memory_order_seq_cst);
          first_time.store(false);
        }
      }

      if (dX.Dim() != X.Dim()*DIM) dX.ReInit(X.Dim()*DIM);
      for (Long i = 0; i < X.Dim(); i++) {
        const Matrix<ValueType> Vi(1, Size(), (Iterator<ValueType>)(ConstIterator<ValueType>)X[i].NodeValues_, false);
        for (Integer k = 0; k < DIM; k++) {
          Matrix<ValueType> Vo(1, Size(), dX[i*DIM+k].NodeValues_, false);
          Matrix<ValueType>::GEMM(Vo, Vi, GradOp[k]);
        }
      }
    }
    static EvalOpType SetupEval(const Matrix<ValueType>& X) {
      Long N = X.Dim(1);
      SCTL_ASSERT(X.Dim(0) == DIM);
      Matrix<ValueType> M(Size(), N);
      { // Set M
        auto nodes = Basis<ValueType,1,ORDER>::Nodes();
        Integer NN = Basis<ValueType,1,ORDER>::Size();
        Matrix<ValueType> M_(NN, DIM*N);
        for (Long i = 0; i < DIM*N; i++) {
          ValueType x = X[0][i];
          for (Integer j = 0; j < NN; j++) {
            ValueType y = 1;
            for (Integer k = 0; k < NN; k++) {
              y *= (j==k ? 1 : (nodes[0][k] - x) / (nodes[0][k] - nodes[0][j]));
            }
            M_[j][i] = y;
          }
        }
        if (DIM == 1) {
          SCTL_ASSERT(M.Dim(0) == M_.Dim(0));
          SCTL_ASSERT(M.Dim(1) == M_.Dim(1));
          M = M_;
        } else {
          Integer NNN = 1;
          M = 1;
          for (Integer d = 0; d < DIM; d++) {
            for (Integer k = 1; k < NN; k++) {
              for (Integer j = 0; j < NNN; j++) {
                for (Long i = 0; i < N; i++) {
                  M[k*NNN+j][i] = M[j][i] * M_[k][d*N+i];
                }
              }
            }
            { // k = 0
              for (Integer j = 0; j < NNN; j++) {
                for (Long i = 0; i < N; i++) {
                  M[j][i] *= M_[0][d*N+i];
                }
              }
            }
            NNN *= NN;
          }
        }
      }
      return M;
    }
    static void Eval(Matrix<ValueType>& Y, const Vector<Basis>& X, const EvalOpType& M) {
      Long N0 = X.Dim();
      Long N1 = M.Dim(1);
      SCTL_ASSERT(M.Dim(0) == Size());
      if (Y.Dim(0) != N0 || Y.Dim(1) != N1) Y.ReInit(N0, N1);
      for (Long i = 0; i < N0; i++) {
        const Matrix<ValueType> X_(1,Size(),(Iterator<ValueType>)(ConstIterator<ValueType>)X[i].NodeValues_,false);
        Matrix<ValueType> Y_(1,N1,Y[i],false);
        Matrix<ValueType>::GEMM(Y_,X_,M);
      }
    }

    Basis operator+(Basis X) const {
      for (Long i = 0; i < Size(); i++) X[i] = (*this)[i] + X[i];
      return X;
    }
    Basis operator-(Basis X) const {
      for (Long i = 0; i < Size(); i++) X[i] = (*this)[i] - X[i];
      return X;
    }
    Basis operator*(Basis X) const {
      for (Long i = 0; i < Size(); i++) X[i] = (*this)[i] * X[i];
      return X;
    }
    Basis operator*(Real a) const {
      Basis X = (*this);
      for (Long i = 0; i < Size(); i++) X[i] *= a;
      return X;
    }
    Basis& operator+=(const Basis& X) {
      for (Long i = 0; i < Size(); i++) (*this)[i] += X[i];
      return *this;
    }
    Basis& operator-=(const Basis& X) {
      for (Long i = 0; i < Size(); i++) (*this)[i] -= X[i];
      return *this;
    }
    Basis& operator*=(const Basis& X) {
      for (Long i = 0; i < Size(); i++) (*this)[i] *= X[i];
      return *this;
    }
    Basis& operator*=(Real a) {
      for (Long i = 0; i < Size(); i++) (*this)[i] *= a;
      return *this;
    }
    Basis& operator=(Real a) {
      for (Long i = 0; i < Size(); i++) (*this)[i] = a;
      return *this;
    }


    const ValueType& operator[](Long i) const {
      SCTL_ASSERT(i < Size());
      return NodeValues_[i];
    }
    ValueType& operator[](Long i) {
      SCTL_ASSERT(i < Size());
      return NodeValues_[i];
    }

  private:
    StaticArray<ValueType,Size()> NodeValues_;
};

template <Integer COORD_DIM, class Basis> class ElemList {
  public:

    using CoordBasis = Basis;
    using CoordType = typename CoordBasis::ValueType;

    static constexpr Integer CoordDim() {
      return COORD_DIM;
    }
    static constexpr Integer ElemDim() {
      return CoordBasis::Dim();
    }

    ElemList(Long Nelem = 0) {
      ReInit(Nelem);
    }
    void ReInit(Long Nelem = 0) {
      Nelem_ = Nelem;
      X_.ReInit(Nelem_ * COORD_DIM);
    }
    void ReInit(const Vector<CoordBasis>& X) {
      Nelem_ = X.Dim() / COORD_DIM;
      SCTL_ASSERT(X.Dim() == Nelem_ * COORD_DIM);
      X_ = X;
    }
    Long NElem() const {
      return Nelem_;
    }

    CoordBasis& operator()(Long elem, Integer dim) {
      SCTL_ASSERT(elem >= 0 && elem < Nelem_);
      SCTL_ASSERT(dim >= 0 && dim < COORD_DIM);
      return X_[elem*COORD_DIM+dim];
    }
    const CoordBasis& operator()(Long elem, Integer dim) const {
      SCTL_ASSERT(elem >= 0 && elem < Nelem_);
      SCTL_ASSERT(dim >= 0 && dim < COORD_DIM);
      return X_[elem*COORD_DIM+dim];
    }
    const Vector<CoordBasis>& ElemVector() const {
      return X_;
    }

  private:
    static_assert(CoordBasis::Dim() <= CoordDim(), "Basis dimension can not be greater than COORD_DIM.");
    Vector<CoordBasis> X_;
    Long Nelem_;

    mutable Vector<CoordBasis> dX_;
};

template <class Real> class Quadrature {

    static Real machine_epsilon() {
      Real eps=1;
      while(eps*(Real)0.5+(Real)1.0>1.0) eps*=0.5;
      return eps;
    }

    template <Integer DIM> static void DuffyQuad(Matrix<Real>& nodes, Vector<Real>& weights, const Vector<Real>& coord, Integer order, Real adapt = -1.0) {
      SCTL_ASSERT(coord.Dim() == DIM);
      static Real eps = machine_epsilon()*16;

      Matrix<Real> qx;
      Vector<Real> qw;
      { // Set qx, qw
        Vector<Real> qx0, qw0;
        ChebBasis<Real>::quad_rule(order, qx0, qw0);

        Integer N = sctl::pow<DIM,Integer>(order);
        qx.ReInit(DIM,N);
        qw.ReInit(N);
        qw[0] = 1;

        Integer N_ = 1;
        for (Integer d = 0; d < DIM; d++) {
          for (Integer j = 0; j < order; j++) {
            for (Integer i = 0; i < N_; i++) {
              for (Integer k = 0; k < d; k++) {
                qx[k][j*N_+i] = qx[k][i];
              }
              qx[d][j*N_+i] = qx0[j];
              qw[j*N_+i] = qw[i];
            }
          }
          for (Integer j = 0; j < order; j++) {
            for (Integer i = 0; i < N_; i++) {
              qw[j*N_+i] *= qw0[j];
            }
          }
          N_ *= order;
        }
      }

      Vector<Real> X;
      { // Set X
        StaticArray<Real,2*DIM+2> X_;
        X_[0] = 0;
        X_[1] = adapt;
        for (Integer i = 0; i < DIM; i++) {
          X_[2*i+2] = sctl::fabs<Real>(coord[i]);
          X_[2*i+3] = sctl::fabs<Real>(coord[i]-1);
        }
        std::sort((Iterator<Real>)X_, (Iterator<Real>)X_+2*DIM+2);

        X.PushBack(std::max<Real>(0, X_[2*DIM]-1));
        for (Integer i = 0; i < 2*DIM+2; i++) {
          if (X[X.Dim()-1] < X_[i]) {
            if (X.Dim())
            X.PushBack(X_[i]);
          }
        }

        /////////////////////////////////////////////////////////////////////////////////////////////////
        Vector<Real> r(1);
        r[0] = X[0];
        for (Integer i = 1; i < X.Dim(); i++) {
          while (r[r.Dim() - 1] > 0.0 && (order*0.5) * r[r.Dim() - 1] < X[i]) r.PushBack((order*0.5) * r[r.Dim() - 1]); // TODO
          r.PushBack(X[i]);
        }
        X = r;
        /////////////////////////////////////////////////////////////////////////////////////////////////
      }

      Vector<Real> nds, wts;
      for (Integer k = 0; k < X.Dim()-1; k++) {
        for (Integer dd = 0; dd < 2*DIM; dd++) {
          Integer d0 = (dd>>1);
          StaticArray<Real,2*DIM> range0, range1;
          { // Set range0, range1
            Integer d1 = (dd%2?1:-1);
            for (Integer d = 0; d < DIM; d++) {
              range0[d*2+0] = std::max<Real>(0,std::min<Real>(1,coord[d] - X[k]  ));
              range0[d*2+1] = std::max<Real>(0,std::min<Real>(1,coord[d] + X[k]  ));
              range1[d*2+0] = std::max<Real>(0,std::min<Real>(1,coord[d] - X[k+1]));
              range1[d*2+1] = std::max<Real>(0,std::min<Real>(1,coord[d] + X[k+1]));
            }
            range0[d0*2+0] = std::max<Real>(0,std::min<Real>(1,coord[d0] + d1*X[k+0]));
            range0[d0*2+1] = std::max<Real>(0,std::min<Real>(1,coord[d0] + d1*X[k+0]));
            range1[d0*2+0] = std::max<Real>(0,std::min<Real>(1,coord[d0] + d1*X[k+1]));
            range1[d0*2+1] = std::max<Real>(0,std::min<Real>(1,coord[d0] + d1*X[k+1]));
          }
          { // if volume(range0, range1) == 0 then continue
            Real v0 = 1, v1 = 1;
            for (Integer d = 0; d < DIM; d++) {
              if (d == d0) {
                v0 *= sctl::fabs<Real>(range0[d*2+0]-range1[d*2+0]);
                v1 *= sctl::fabs<Real>(range0[d*2+0]-range1[d*2+0]);
              } else {
                v0 *= range0[d*2+1]-range0[d*2+0];
                v1 *= range1[d*2+1]-range1[d*2+0];
              }
            }
            if (v0 < eps && v1 < eps) continue;
          }
          for (Integer i = 0; i < qx.Dim(1); i++) { // Set nds, wts
            Real w = qw[i];
            Real z = qx[d0][i];
            for (Integer d = 0; d < DIM; d++) {
              Real y = qx[d][i];
              nds.PushBack((range0[d*2+0]*(1-y) + range0[d*2+1]*y)*(1-z) + (range1[d*2+0]*(1-y) + range1[d*2+1]*y)*z);
              if (d == d0) {
                w *= abs(range1[d*2+0] - range0[d*2+0]);
              } else {
                w *= (range0[d*2+1] - range0[d*2+0])*(1-z) + (range1[d*2+1] - range1[d*2+0])*z;
              }
            }
            wts.PushBack(w);
          }
        }
      }
      nodes = Matrix<Real>(nds.Dim()/DIM,DIM,nds.begin()).Transpose();
      weights = wts;
    }

    template <Integer DIM> static void TensorProductGaussQuad(Matrix<Real>& nodes, Vector<Real>& weights, Integer order) {
      Vector<Real> coord(DIM);
      coord = 0;
      coord[0] = -10;
      DuffyQuad<DIM>(nodes, weights, coord, order);
    }



    template <class DensityBasis, class ElemList, class Kernel> static void SetupSingular(Matrix<Real>& M_singular, const Matrix<Real>& trg_nds, const ElemList& elem_lst, const Kernel& kernel, Integer order_singular = 10, Integer order_direct = 10, Real Rqbx = 0) {
      using CoordBasis = typename ElemList::CoordBasis;
      using CoordEvalOpType = typename CoordBasis::EvalOpType;
      using DensityEvalOpType = typename DensityBasis::EvalOpType;

      constexpr Integer CoordDim = ElemList::CoordDim();
      constexpr Integer ElemDim = ElemList::ElemDim();
      constexpr Integer KDIM0 = Kernel::SrcDim();
      constexpr Integer KDIM1 = Kernel::TrgDim();

      const Long Nelem = elem_lst.NElem();
      const Integer Ntrg = trg_nds.Dim(1);
      SCTL_ASSERT(trg_nds.Dim(0) == ElemDim);

      const Vector<CoordBasis>& X = elem_lst.ElemVector();
      Vector<CoordBasis> dX;
      CoordBasis::Grad(dX, X);

      Vector<Real> Xt, Xnt;
      { // Set Xt, Xnt
        auto Meval = CoordBasis::SetupEval(trg_nds);
        eval_basis(Xt, X, CoordDim, trg_nds.Dim(1), Meval);

        Xnt = Xt;
        Vector<Real> dX_;
        eval_basis(dX_, dX, 2*CoordDim, trg_nds.Dim(1), Meval);

        for (Long i = 0; i < Ntrg; i++) {
          for (Long j = 0; j < Nelem; j++) {
            auto Xn = Xnt.begin() + (j*Ntrg+i)*CoordDim;
            auto dX0 = dX_.begin() + (j*Ntrg+i)*2*CoordDim;

            StaticArray<Real,CoordDim> normal;
            normal[0] = dX0[2]*dX0[5] - dX0[4]*dX0[3];
            normal[1] = dX0[4]*dX0[1] - dX0[0]*dX0[5];
            normal[2] = dX0[0]*dX0[3] - dX0[2]*dX0[1];
            Real Xa = sctl::sqrt<Real>(normal[0]*normal[0]+normal[1]*normal[1]+normal[2]*normal[2]);
            Real invXa = 1/Xa;
            normal[0] *= invXa;
            normal[1] *= invXa;
            normal[2] *= invXa;

            Real sqrt_Xa = sqrt<Real>(Xa);
            Xn[0] = normal[0]*sqrt_Xa*Rqbx;
            Xn[1] = normal[1]*sqrt_Xa*Rqbx;
            Xn[2] = normal[2]*sqrt_Xa*Rqbx;
          }
        }
      }
      SCTL_ASSERT(Xt.Dim() == Nelem * Ntrg * CoordDim);

      auto& M = M_singular;
      M.ReInit(Nelem * KDIM0 * DensityBasis::Size(), KDIM1 * Ntrg);
      #pragma omp parallel for schedule(static)
      for (Long i = 0; i < Ntrg; i++) { // Set M (singular)
        Matrix<Real> quad_nds;
        Vector<Real> quad_wts;
        { // Set quad_nds, quad_wts
          StaticArray<Real,ElemDim> trg_node_;
          for (Integer k = 0; k < ElemDim; k++) {
            trg_node_[k] = trg_nds[k][i];
          }
          Vector<Real> trg_node(ElemDim, trg_node_, false);
          DuffyQuad<ElemDim>(quad_nds, quad_wts, trg_node, order_singular, fabs(Rqbx));
        }
        const CoordEvalOpType CoordEvalOp = CoordBasis::SetupEval(quad_nds);
        Integer Nnds = quad_wts.Dim();

        Vector<Real> X_, dX_, Xa_, Xn_;
        { // Set X_, dX_
          eval_basis(X_, X, CoordDim, Nnds, CoordEvalOp);
          eval_basis(dX_, dX, CoordDim * ElemDim, Nnds, CoordEvalOp);
        }
        if (CoordDim == 3 && ElemDim == 2) { // Compute Xa_, Xn_
          Long N = Nelem*Nnds;
          Xa_.ReInit(N);
          Xn_.ReInit(N*CoordDim);
          for (Long j = 0; j < N; j++) {
            StaticArray<Real,CoordDim> normal;
            normal[0] = dX_[j*6+2]*dX_[j*6+5] - dX_[j*6+4]*dX_[j*6+3];
            normal[1] = dX_[j*6+4]*dX_[j*6+1] - dX_[j*6+0]*dX_[j*6+5];
            normal[2] = dX_[j*6+0]*dX_[j*6+3] - dX_[j*6+2]*dX_[j*6+1];
            Xa_[j] = sctl::sqrt<Real>(normal[0]*normal[0]+normal[1]*normal[1]+normal[2]*normal[2]);
            Real invXa = 1/Xa_[j];
            Xn_[j*3+0] = normal[0] * invXa;
            Xn_[j*3+1] = normal[1] * invXa;
            Xn_[j*3+2] = normal[2] * invXa;
          }
        }

        DensityEvalOpType DensityEvalOp;
        if (std::is_same<CoordBasis,DensityBasis>::value) {
          DensityEvalOp = CoordEvalOp;
        } else {
          DensityEvalOp = DensityBasis::SetupEval(quad_nds);
        }

        for (Long j = 0; j < Nelem; j++) {
          Matrix<Real> M__(Nnds * KDIM0, KDIM1);
          if (Rqbx == 0) { // Set kernel matrix M__
            const Vector<Real> X0_(CoordDim, Xt.begin() + (j * Ntrg + i) * CoordDim, false);
            const Vector<Real> X__(Nnds * CoordDim, X_.begin() + j * Nnds * CoordDim, false);
            const Vector<Real> Xn__(Nnds * CoordDim, Xn_.begin() + j * Nnds * CoordDim, false);
            kernel.template KernelMatrix<Real>(M__, X0_, X__, Xn__);
          } else {
            Vector<Real> X0_(CoordDim);
            constexpr Integer qbx_order = 6;
            StaticArray<Matrix<Real>,qbx_order> M___;
            for (Integer k = 0; k < qbx_order; k++) { // Set kernel matrix M___
              for (Integer kk = 0; kk < CoordDim; kk++) X0_[kk] = Xt[(j * Ntrg + i) * CoordDim + kk] + (k+1) * Xnt[(j * Ntrg + i) * CoordDim + kk];
              const Vector<Real> X__(Nnds * CoordDim, X_.begin() + j * Nnds * CoordDim, false);
              const Vector<Real> Xn__(Nnds * CoordDim, Xn_.begin() + j * Nnds * CoordDim, false);
              kernel.template KernelMatrix<Real>(M___[k], X0_, X__, Xn__);
            }

            for (Long k = 0; k < Nnds * KDIM0 * KDIM1; k++) {
              M__[0][k] = 0;
              M__[0][k] +=   6*M___[0][0][k];
              M__[0][k] += -15*M___[1][0][k];
              M__[0][k] +=  20*M___[2][0][k];
              M__[0][k] += -15*M___[3][0][k];
              M__[0][k] +=   6*M___[4][0][k];
              M__[0][k] +=  -1*M___[5][0][k];
            }
          }

          for (Long k0 = 0; k0 < KDIM0; k0++) {
            for (Long k1 = 0; k1 < KDIM1; k1++) {
              for (Long l = 0; l < DensityBasis::Size(); l++) {
                Real M_lk = 0;
                for (Long n = 0; n < Nnds; n++) {
                  Real quad_wt = Xa_[j * Nnds + n] * quad_wts[n];
                  M_lk += DensityEvalOp[l][n] * quad_wt * M__[n*KDIM0+k0][k1];
                }
                M[(j * KDIM0 + k0) * DensityBasis::Size() + l][k1 * Ntrg + i] = M_lk;
              }
            }
          }
        }
      }
      { // Set M (subtract direct)
        Matrix<Real> quad_nds;
        Vector<Real> quad_wts;
        TensorProductGaussQuad<ElemDim>(quad_nds, quad_wts, order_direct);
        const CoordEvalOpType CoordEvalOp = CoordBasis::SetupEval(quad_nds);
        Integer Nnds = quad_wts.Dim();

        Vector<Real> X_, dX_, Xa_, Xn_;
        { // Set X_, dX_
          eval_basis(X_, X, CoordDim, Nnds, CoordEvalOp);
          eval_basis(dX_, dX, CoordDim * ElemDim, Nnds, CoordEvalOp);
        }
        if (CoordDim == 3 && ElemDim == 2) { // Compute Xa_, Xn_
          Long N = Nelem*Nnds;
          Xa_.ReInit(N);
          Xn_.ReInit(N*CoordDim);
          for (Long j = 0; j < N; j++) {
            StaticArray<Real,CoordDim> normal;
            normal[0] = dX_[j*6+2]*dX_[j*6+5] - dX_[j*6+4]*dX_[j*6+3];
            normal[1] = dX_[j*6+4]*dX_[j*6+1] - dX_[j*6+0]*dX_[j*6+5];
            normal[2] = dX_[j*6+0]*dX_[j*6+3] - dX_[j*6+2]*dX_[j*6+1];
            Xa_[j] = sctl::sqrt<Real>(normal[0]*normal[0]+normal[1]*normal[1]+normal[2]*normal[2]);
            Real invXa = 1/Xa_[j];
            Xn_[j*3+0] = normal[0] * invXa;
            Xn_[j*3+1] = normal[1] * invXa;
            Xn_[j*3+2] = normal[2] * invXa;
          }
        }

        DensityEvalOpType DensityEvalOp;
        if (std::is_same<CoordBasis,DensityBasis>::value) {
          DensityEvalOp = CoordEvalOp;
        } else {
          DensityEvalOp = DensityBasis::SetupEval(quad_nds);
        }

        #pragma omp parallel for schedule(static)
        for (Long i = 0; i < Ntrg; i++) { // Subtract direct contribution
          for (Long j = 0; j < Nelem; j++) {
            Matrix<Real> M__(Nnds * KDIM0, KDIM1);
            { // Set kernel matrix M__
              const Vector<Real> X0_(CoordDim, (Iterator<Real>)Xt.begin() + (j * Ntrg + i) * CoordDim, false);
              const Vector<Real> X__(Nnds * CoordDim, X_.begin() + j * Nnds * CoordDim, false);
              const Vector<Real> Xn__(Nnds * CoordDim, Xn_.begin() + j * Nnds * CoordDim, false);
              kernel.template KernelMatrix<Real>(M__, X0_, X__, Xn__);
            }
            for (Long k0 = 0; k0 < KDIM0; k0++) {
              for (Long k1 = 0; k1 < KDIM1; k1++) {
                for (Long l = 0; l < DensityBasis::Size(); l++) {
                  Real M_lk = 0;
                  for (Long n = 0; n < Nnds; n++) {
                    Real quad_wt = Xa_[j * Nnds + n] * quad_wts[n];
                    M_lk += DensityEvalOp[l][n] * quad_wt * M__[n*KDIM0+k0][k1];
                  }
                  M[(j * KDIM0 + k0) * DensityBasis::Size() + l][k1 * Ntrg + i] -= M_lk;
                }
              }
            }
          }
        }
      }
    }

    template <class DensityBasis> static void EvalSingular(Matrix<Real>& U, const Vector<DensityBasis>& density, const Matrix<Real>& M, Integer KDIM0_, Integer KDIM1_) {
      if (M.Dim(0) == 0 || M.Dim(1) == 0) {
        U.ReInit(0,0);
        return;
      }

      const Long Ntrg = M.Dim(1) / KDIM1_;
      SCTL_ASSERT(M.Dim(1) == KDIM1_ * Ntrg);

      const Long Nelem = M.Dim(0) / (KDIM0_ * DensityBasis::Size());
      SCTL_ASSERT(M.Dim(0) == Nelem * KDIM0_ * DensityBasis::Size());

      const Integer dof = density.Dim() / (Nelem * KDIM0_);
      SCTL_ASSERT(density.Dim() == Nelem * dof * KDIM0_);

      if (U.Dim(0) != Nelem * dof * KDIM1_ || U.Dim(1) != Ntrg) {
        U.ReInit(Nelem * dof * KDIM1_, Ntrg);
        U = 0;
      }
      for (Long j = 0; j < Nelem; j++) {
        const Matrix<Real> M_(KDIM0_ * DensityBasis::Size(), KDIM1_ * Ntrg, (Iterator<Real>)M[j * KDIM0_ * DensityBasis::Size()], false);
        Matrix<Real> U_(dof, KDIM1_ * Ntrg, U[j*dof*KDIM1_], false);
        Matrix<Real> F_(dof, KDIM0_ * DensityBasis::Size());
        for (Long i = 0; i < dof; i++) {
          for (Long k = 0; k < KDIM0_; k++) {
            for (Long l = 0; l < DensityBasis::Size(); l++) {
              F_[i][k * DensityBasis::Size() + l] = density[(j * dof + i) * KDIM0_ + k][l];
            }
          }
        }
        Matrix<Real>::GEMM(U_, F_, M_);
      }
    }



    template <Integer DIM> struct PointData {
      bool operator<(const PointData& p) const {
        return mid < p.mid;
      }

      Long rank;
      Long surf_rank;
      Morton<DIM> mid;
      StaticArray<Real,DIM> coord;
      Real radius2;
    };

    template <class T1, class T2> struct Pair {
      Pair() {}

      Pair(T1 x, T2 y) : first(x), second(y) {}

      bool operator<(const Pair& p) const {
        return (first < p.first) || (((first == p.first) && (second < p.second)));
      }

      T1 first;
      T2 second;
    };

    template <class ElemList> static void BuildNbrList(Vector<Pair<Long,Long>>& pair_lst, const Vector<Real>& Xt, const Vector<Long>& trg_surf, const ElemList& elem_lst, Real distance_factor, Real period_length, const Comm& comm) {
      using CoordBasis = typename ElemList::CoordBasis;
      constexpr Integer CoordDim = ElemList::CoordDim();
      constexpr Integer ElemDim = ElemList::ElemDim();
      using PtData = PointData<CoordDim>;
      const Integer rank = comm.Rank();

      Real R0 = 0;
      StaticArray<Real,CoordDim> X0;
      { // Find bounding box
        Long N = Xt.Dim() / CoordDim;
        SCTL_ASSERT(Xt.Dim() == N * CoordDim);
        SCTL_ASSERT(N);

        StaticArray<Real,CoordDim*2> Xloc;
        StaticArray<Real,CoordDim*2> Xglb;
        for (Integer k = 0; k < CoordDim; k++) {
          Xloc[0*CoordDim+k] = Xt[k];
          Xloc[1*CoordDim+k] = Xt[k];
        }
        for (Long i = 0; i < N; i++) {
          for (Integer k = 0; k < CoordDim; k++) {
            Xloc[0*CoordDim+k] = std::min<Real>(Xloc[0*CoordDim+k], Xt[i*CoordDim+k]);
            Xloc[1*CoordDim+k] = std::max<Real>(Xloc[1*CoordDim+k], Xt[i*CoordDim+k]);
          }
        }
        comm.Allreduce((ConstIterator<Real>)Xloc+0*CoordDim, (Iterator<Real>)Xglb+0*CoordDim, CoordDim, Comm::CommOp::MIN);
        comm.Allreduce((ConstIterator<Real>)Xloc+1*CoordDim, (Iterator<Real>)Xglb+1*CoordDim, CoordDim, Comm::CommOp::MAX);
        for (Integer k = 0; k < CoordDim; k++) {
          R0 = std::max(R0, Xglb[1*CoordDim+k]-Xglb[0*CoordDim+k]);
        }

        R0 = R0 * 2.0;
        for (Integer k = 0; k < CoordDim; k++) {
          X0[k] = Xglb[k] - R0*0.25;
        }
      }
      if (period_length > 0) {
        R0 = period_length;
      }

      Vector<PtData> PtSrc, PtTrg;
      Integer order_upsample = (Integer)(const_pi<Real>() / distance_factor + 0.5);
      { // Set PtSrc
        const Vector<CoordBasis>& X_elem_lst = elem_lst.ElemVector();
        Vector<CoordBasis> dX_elem_lst;
        CoordBasis::Grad(dX_elem_lst, X_elem_lst);

        Matrix<Real> nds;
        Vector<Real> wts;
        TensorProductGaussQuad<ElemDim>(nds, wts, order_upsample);
        const Long Nnds = nds.Dim(1);

        Vector<Real> X, dX;
        const auto CoordEvalOp = CoordBasis::SetupEval(nds);
        eval_basis(X, X_elem_lst, CoordDim, Nnds, CoordEvalOp);
        eval_basis(dX, dX_elem_lst, CoordDim * ElemDim, Nnds, CoordEvalOp);

        const Long N = X.Dim() / CoordDim;
        const Long Nelem = elem_lst.NElem();
        SCTL_ASSERT(X.Dim() == N * CoordDim);
        SCTL_ASSERT(N == Nelem * Nnds);

        Long rank_offset, surf_rank_offset;
        { // Set rank_offset, surf_rank_offset
          comm.Scan(Ptr2ConstItr<Long>(&N,1), Ptr2Itr<Long>(&rank_offset,1), 1, Comm::CommOp::SUM);
          comm.Scan(Ptr2ConstItr<Long>(&Nelem,1), Ptr2Itr<Long>(&surf_rank_offset,1), 1, Comm::CommOp::SUM);
          surf_rank_offset -= Nelem;
          rank_offset -= N;
        }

        PtSrc.ReInit(N);
        const Real R0inv = 1.0 / R0;
        for (Long i = 0; i < N; i++) { // Set coord
          for (Integer k = 0; k < CoordDim; k++) {
            PtSrc[i].coord[k] = (X[i*CoordDim+k] - X0[k]) * R0inv;
          }
        }
        if (period_length > 0) { // Wrap-around coord
          for (Long i = 0; i < N; i++) {
            auto& x = PtSrc[i].coord;
            for (Integer k = 0; k < CoordDim; k++) {
              x[k] -= (Long)(x[k]);
            }
          }
        }
        for (Long i = 0; i < N; i++) { // Set radius2, mid, rank
          Integer depth = 0;
          { // Set radius2, depth
            Real radius2 = 0;
            for (Integer k0 = 0; k0 < ElemDim; k0++) {
              Real R2 = 0;
              for (Integer k1 = 0; k1 < CoordDim; k1++) {
                Real dX_ = dX[(i*CoordDim+k1)*ElemDim+k0];
                R2 += dX_*dX_;
              }
              radius2 = std::max(radius2, R2);
            }
            radius2 *= R0inv*R0inv * distance_factor*distance_factor;
            PtSrc[i].radius2 = radius2;

            Long Rinv = (Long)(1.0/radius2);
            while (Rinv > 0) {
              Rinv = (Rinv>>2);
              depth++;
            }
          }
          PtSrc[i].mid = Morton<CoordDim>((Iterator<Real>)PtSrc[i].coord, std::min(Morton<CoordDim>::MaxDepth(),depth));
          PtSrc[i].rank = rank_offset + i;
        }
        for (Long i = 0 ; i < Nelem; i++) { // Set surf_rank
          for (Long j = 0; j < Nnds; j++) {
            PtSrc[i*Nnds+j].surf_rank = surf_rank_offset + i;
          }
        }

        Vector<PtData> PtSrcSorted;
        comm.HyperQuickSort(PtSrc, PtSrcSorted);
        PtSrc.Swap(PtSrcSorted);
      }
      { // Set PtTrg
        const Long N = Xt.Dim() / CoordDim;
        SCTL_ASSERT(Xt.Dim() == N * CoordDim);

        Long rank_offset;
        { // Set rank_offset
          comm.Scan(Ptr2ConstItr<Long>(&N,1), Ptr2Itr<Long>(&rank_offset,1), 1, Comm::CommOp::SUM);
          rank_offset -= N;
        }

        PtTrg.ReInit(N);
        const Real R0inv = 1.0 / R0;
        for (Long i = 0; i < N; i++) { // Set coord
          for (Integer k = 0; k < CoordDim; k++) {
            PtTrg[i].coord[k] = (Xt[i*CoordDim+k] - X0[k]) * R0inv;
          }
        }
        if (period_length > 0) { // Wrap-around coord
          for (Long i = 0; i < N; i++) {
            auto& x = PtTrg[i].coord;
            for (Integer k = 0; k < CoordDim; k++) {
              x[k] -= (Long)(x[k]);
            }
          }
        }
        for (Long i = 0; i < N; i++) { // Set radius2, mid, rank
          PtTrg[i].radius2 = 0;
          PtTrg[i].mid = Morton<CoordDim>((Iterator<Real>)PtTrg[i].coord);
          PtTrg[i].rank = rank_offset + i;
        }
        if (trg_surf.Dim()) { // Set surf_rank
          SCTL_ASSERT(trg_surf.Dim() == N);
          for (Long i = 0; i < N; i++) {
            PtTrg[i].surf_rank = trg_surf[i];
          }
        } else {
          for (Long i = 0; i < N; i++) {
            PtTrg[i].surf_rank = -1;
          }
        }

        Vector<PtData> PtTrgSorted;
        comm.HyperQuickSort(PtTrg, PtTrgSorted);
        PtTrg.Swap(PtTrgSorted);
      }

      Tree<CoordDim> tree(comm);
      { // Init tree
        Vector<Real> Xall(PtSrc.Dim()+PtTrg.Dim());
        { // Set Xall
          Xall.ReInit((PtSrc.Dim()+PtTrg.Dim())*CoordDim);
          Long Nsrc = PtSrc.Dim();
          Long Ntrg = PtTrg.Dim();
          for (Long i = 0; i < Nsrc; i++) {
            for (Integer k = 0; k < CoordDim; k++) {
              Xall[i*CoordDim+k] = PtSrc[i].coord[k];
            }
          }
          for (Long i = 0; i < Ntrg; i++) {
            for (Integer k = 0; k < CoordDim; k++) {
              Xall[(Nsrc+i)*CoordDim+k] = PtTrg[i].coord[k];
            }
          }
        }
        tree.UpdateRefinement(Xall, 1000, true, period_length>0);
      }
      { // Repartition PtSrc, PtTrg
        PtData splitter;
        splitter.mid = tree.GetPartitionMID()[rank];
        comm.PartitionS(PtSrc, splitter);
        comm.PartitionS(PtTrg, splitter);
      }
      { // Add tree data PtSrc
        const auto& node_mid = tree.GetNodeMID();
        const Long N = node_mid.Dim();
        SCTL_ASSERT(N);

        Vector<Long> dsp(N), cnt(N);
        for (Long i = 0; i < N; i++) {
          PtData m0;
          m0.mid = node_mid[i];
          dsp[i] = std::lower_bound(PtSrc.begin(), PtSrc.end(), m0) - PtSrc.begin();
        }
        for (Long i = 0; i < N-1; i++) {
          cnt[i] = dsp[i+1] - dsp[i];
        }
        cnt[N-1] = PtSrc.Dim() - dsp[N-1];
        tree.AddData("PtSrc", PtSrc, cnt);
      }
      tree.template Broadcast<PtData>("PtSrc");

      { // Build pair_lst
        Vector<Long> cnt;
        Vector<PtData> PtSrc;
        tree.GetData(PtSrc, cnt, "PtSrc");
        const auto& node_mid = tree.GetNodeMID();
        const auto& node_attr = tree.GetNodeAttr();

        Vector<Morton<CoordDim>> nbr_mid_tmp;
        for (Long i = 0; i < node_mid.Dim(); i++) {
          if (node_attr[i].Leaf && !node_attr[i].Ghost) {
            Vector<Morton<CoordDim>> child_mid;
            node_mid[i].Children(child_mid);
            for (const auto& trg_mid : child_mid) {
              Integer d0 = trg_mid.Depth();
              Vector<PtData> Src, Trg;
              { // Set Trg
                PtData m0, m1;
                m0.mid = trg_mid;
                m1.mid = trg_mid.Next();
                Long a = std::lower_bound(PtTrg.begin(), PtTrg.end(), m0) - PtTrg.begin();
                Long b = std::lower_bound(PtTrg.begin(), PtTrg.end(), m1) - PtTrg.begin();
                Trg.ReInit(b-a, PtTrg.begin()+a, false);
                if (!Trg.Dim()) continue;
              }

              Vector<std::set<Long>> near_elem(Trg.Dim());
              for (Integer d = 0; d <= d0; d++) {
                trg_mid.NbrList(nbr_mid_tmp, d, period_length>0);
                for (const auto& src_mid : nbr_mid_tmp) { // Set Src
                  PtData m0, m1;
                  m0.mid = src_mid;
                  m1.mid = (d==d0 ? src_mid.Next() : src_mid.Ancestor(d+1));
                  Long a = std::lower_bound(PtSrc.begin(), PtSrc.end(), m0) - PtSrc.begin();
                  Long b = std::lower_bound(PtSrc.begin(), PtSrc.end(), m1) - PtSrc.begin();
                  Src.ReInit(b-a, PtSrc.begin()+a, false);
                  if (!Src.Dim()) continue;

                  for (Long t = 0; t < Trg.Dim(); t++) { // set near_elem[t] <-- {s : dist(s,t) < radius(s)}
                    for (Long s = 0; s < Src.Dim(); s++) {
                      if (Trg[t].surf_rank != Src[s].surf_rank) {
                        Real R2 = 0;
                        for (Integer k = 0; k < CoordDim; k++) {
                          Real dx = (Src[s].coord[k] - Trg[t].coord[k]);
                          R2 += dx * dx;
                        }
                        if (R2 < Src[s].radius2) {
                          near_elem[t].insert(Src[s].surf_rank);
                        }
                      }
                    }
                  }
                }
              }

              for (Long t = 0; t < Trg.Dim(); t++) { // Set pair_lst
                for (Long elem_idx : near_elem[t]) {
                  pair_lst.PushBack(Pair<Long,Long>(elem_idx,Trg[t].rank));
                }
              }
            }
          }
        }
      }
      { // Sort and repartition pair_lst
        Vector<Pair<Long,Long>> pair_lst_sorted;
        comm.HyperQuickSort(pair_lst, pair_lst_sorted);

        Long surf_rank_offset;
        const Long Nelem = elem_lst.NElem();
        comm.Scan(Ptr2ConstItr<Long>(&Nelem,1), Ptr2Itr<Long>(&surf_rank_offset,1), 1, Comm::CommOp::SUM);
        surf_rank_offset -= Nelem;

        comm.PartitionS(pair_lst_sorted, Pair<Long,Long>(surf_rank_offset,0));
        pair_lst.Swap(pair_lst_sorted);
      }
    }

    template <class ElemList> static void BuildNbrListDeprecated(Vector<Pair<Long,Long>>& pair_lst, const Vector<Real>& Xt, const ElemList& elem_lst, const Matrix<Real>& surf_nds, Real distance_factor) {
      using CoordBasis = typename ElemList::CoordBasis;
      constexpr Integer CoordDim = ElemList::CoordDim();
      constexpr Integer ElemDim = ElemList::ElemDim();
      const Long Nelem = elem_lst.NElem();

      const Long Ntrg = Xt.Dim() / CoordDim;
      SCTL_ASSERT(Xt.Dim() == Ntrg * CoordDim);

      Long Nnds, Nsurf_nds;
      Vector<Real> X_surf, X, dX;
      Integer order_upsample = (Integer)(const_pi<Real>() / distance_factor + 0.5);
      { // Set X, dX
        const Vector<CoordBasis>& X_elem_lst = elem_lst.ElemVector();
        Vector<CoordBasis> dX_elem_lst;
        CoordBasis::Grad(dX_elem_lst, X_elem_lst);

        Matrix<Real> nds_upsample;
        Vector<Real> wts_upsample;
        TensorProductGaussQuad<ElemDim>(nds_upsample, wts_upsample, order_upsample);

        Nnds = nds_upsample.Dim(1);
        const auto CoordEvalOp = CoordBasis::SetupEval(nds_upsample);
        eval_basis(X, X_elem_lst, CoordDim, nds_upsample.Dim(1), CoordEvalOp);
        eval_basis(dX, dX_elem_lst, CoordDim * ElemDim, nds_upsample.Dim(1), CoordEvalOp);

        Nsurf_nds = surf_nds.Dim(1);
        const auto CoordEvalOp_surf = CoordBasis::SetupEval(surf_nds);
        eval_basis(X_surf, X_elem_lst, CoordDim, Nsurf_nds, CoordEvalOp_surf);
      }

      Real d2 = distance_factor * distance_factor;
      for (Long i = 0; i < Nelem; i++) {
        std::set<Long> near_pts;
        std::set<Long> self_pts;
        for (Long j = 0; j < Nnds; j++) {
          Real R2_max = 0;
          StaticArray<Real, CoordDim> X0;
          for (Integer k = 0; k < CoordDim; k++) {
            X0[k] = X[(i*Nnds+j)*CoordDim+k];
          }
          for (Integer k0 = 0; k0 < ElemDim; k0++) {
            Real R2 = 0;
            for (Integer k1 = 0; k1 < CoordDim; k1++) {
              Real dX_ = dX[((i*Nnds+j)*CoordDim+k1)*ElemDim+k0];
              R2 += dX_*dX_;
            }
            R2_max = std::max(R2_max, R2*d2);
          }

          for (Long k = 0; k < Ntrg; k++) {
            Real R2 = 0;
            for (Integer l = 0; l < CoordDim; l++) {
              Real dX = Xt[k*CoordDim+l]- X0[l];
              R2 += dX * dX;
            }
            if (R2 < R2_max) near_pts.insert(k);
          }
        }
        for (Long j = 0; j < Nsurf_nds; j++) {
          StaticArray<Real, CoordDim> X0;
          for (Integer k = 0; k < CoordDim; k++) {
            X0[k] = X_surf[(i*Nsurf_nds+j)*CoordDim+k];
          }
          for (Long k = 0; k < Ntrg; k++) {
            Real R2 = 0;
            for (Integer l = 0; l < CoordDim; l++) {
              Real dX = Xt[k*CoordDim+l]- X0[l];
              R2 += dX * dX;
            }
            if (R2 == 0) self_pts.insert(k);
          }
        }
        for (Long trg_idx : self_pts) {
          near_pts.erase(trg_idx);
        }
        for (Long trg_idx : near_pts) {
          pair_lst.PushBack(Pair<Long,Long>(i,trg_idx));
        }
      }
    }

    template <class DensityBasis, class ElemList, class Kernel> static void SetupNearSingular(Matrix<Real>& M_near_singular, Vector<Pair<Long,Long>>& pair_lst, const Vector<Real>& Xt_, const Vector<Long>& trg_surf, const ElemList& elem_lst, const Kernel& kernel, Integer order_singular, Integer order_direct, Real period_length, const Comm& comm) {
      static_assert(std::is_same<Real,typename DensityBasis::ValueType>::value);
      static_assert(std::is_same<Real,typename ElemList::CoordType>::value);
      static_assert(DensityBasis::Dim() == ElemList::ElemDim());
      using CoordBasis = typename ElemList::CoordBasis;
      using CoordEvalOpType = typename CoordBasis::EvalOpType;
      using DensityEvalOpType = typename DensityBasis::EvalOpType;

      constexpr Integer CoordDim = ElemList::CoordDim();
      constexpr Integer ElemDim = ElemList::ElemDim();
      constexpr Integer KDIM0 = Kernel::SrcDim();
      constexpr Integer KDIM1 = Kernel::TrgDim();
      const Long Nelem = elem_lst.NElem();

      BuildNbrList(pair_lst, Xt_, trg_surf, elem_lst, 2.5/order_direct, period_length, comm);
      const Long Ninterac = pair_lst.Dim();

      Vector<Real> Xt;
      { // Set Xt
        Integer rank = comm.Rank();
        Integer np = comm.Size();

        Vector<Long> splitter_ranks;
        { // Set splitter_ranks
          Vector<Long> cnt(np);
          const Long N = Xt_.Dim() / CoordDim;
          comm.Allgather(Ptr2ConstItr<Long>(&N,1), 1, cnt.begin(), 1);
          scan(splitter_ranks, cnt);
        }

        Vector<Long> scatter_index, recv_index, recv_cnt(np), recv_dsp(np);
        { // Set scatter_index, recv_index, recv_cnt, recv_dsp
          { // Set scatter_index, recv_index
            Vector<Pair<Long,Long>> scatter_pair(pair_lst.Dim());
            for (Long i = 0; i < pair_lst.Dim(); i++) {
              scatter_pair[i] = Pair<Long,Long>(pair_lst[i].second,i);
            }
            omp_par::merge_sort(scatter_pair.begin(), scatter_pair.end());

            recv_index.ReInit(scatter_pair.Dim());
            scatter_index.ReInit(scatter_pair.Dim());
            for (Long i = 0; i < scatter_index.Dim(); i++) {
              recv_index[i] = scatter_pair[i].first;
              scatter_index[i] = scatter_pair[i].second;
            }
          }
          for (Integer i = 0; i < np; i++) {
            recv_dsp[i] = std::lower_bound(recv_index.begin(), recv_index.end(), splitter_ranks[i]) - recv_index.begin();
          }
          for (Integer i = 0; i < np-1; i++) {
            recv_cnt[i] = recv_dsp[i+1] - recv_dsp[i];
          }
          recv_cnt[np-1] = recv_index.Dim() - recv_dsp[np-1];
        }

        Vector<Long> send_index, send_cnt(np), send_dsp(np);
        { // Set send_index, send_cnt, send_dsp
          comm.Alltoall(recv_cnt.begin(), 1, send_cnt.begin(), 1);
          scan(send_dsp, send_cnt);
          send_index.ReInit(send_cnt[np-1] + send_dsp[np-1]);
          comm.Alltoallv(recv_index.begin(), recv_cnt.begin(), recv_dsp.begin(), send_index.begin(), send_cnt.begin(), send_dsp.begin());
        }

        Vector<Real> Xt_send(send_index.Dim() * CoordDim);
        for (Long i = 0; i < send_index.Dim(); i++) { // Set Xt_send
          Long idx = send_index[i] - splitter_ranks[rank];
          for (Integer k = 0; k < CoordDim; k++) {
            Xt_send[i*CoordDim+k] = Xt_[idx*CoordDim+k];
          }
        }

        Vector<Real> Xt_recv(recv_index.Dim() * CoordDim);
        { // Set Xt_recv
          for (Long i = 0; i < np; i++) {
            send_cnt[i] *= CoordDim;
            send_dsp[i] *= CoordDim;
            recv_cnt[i] *= CoordDim;
            recv_dsp[i] *= CoordDim;
          }
          comm.Alltoallv(Xt_send.begin(), send_cnt.begin(), send_dsp.begin(), Xt_recv.begin(), recv_cnt.begin(), recv_dsp.begin());
        }

        Xt.ReInit(scatter_index.Dim() * CoordDim);
        for (Long i = 0; i < scatter_index.Dim(); i++) { // Set Xt
          Long idx = scatter_index[i];
          for (Integer k = 0; k < CoordDim; k++) {
            Xt[idx*CoordDim+k] = Xt_recv[i*CoordDim+k];
          }
        }
      }

      const Vector<CoordBasis>& X = elem_lst.ElemVector();
      Vector<CoordBasis> dX;
      CoordBasis::Grad(dX, X);

      Long elem_rank_offset;
      { // Set elem_rank_offset
        comm.Scan(Ptr2ConstItr<Long>(&Nelem,1), Ptr2Itr<Long>(&elem_rank_offset,1), 1, Comm::CommOp::SUM);
        elem_rank_offset -= Nelem;
      }

      auto& M = M_near_singular;
      M.ReInit(Ninterac * KDIM0 * DensityBasis::Size(), KDIM1);
      #pragma omp parallel for schedule(static)
      for (Long j = 0; j < Ninterac; j++) { // Set M (near-singular)
        const Long src_idx = pair_lst[j].first - elem_rank_offset;

        Real adapt = -1.0;
        Tensor<Real,true,ElemDim,1> u0;
        { // Set u0 (project target point to the surface patch in parameter space)
          ConstIterator<Real> Xt_ = Xt.begin() + j * CoordDim;
          const auto& nodes = CoordBasis::Nodes();

          Long min_idx = -1;
          Real min_R2 = 1e10;
          for (Long i = 0; i < CoordBasis::Size(); i++) {
            Real R2 = 0;
            for (Integer k = 0; k < CoordDim; k++) {
              Real dX = X[src_idx * CoordDim + k][i] - Xt_[k];
              R2 += dX * dX;
            }
            if (R2 < min_R2) {
              min_R2 = R2;
              min_idx = i;
            }
          }
          SCTL_ASSERT(min_idx >= 0);
          for (Integer k = 0; k < ElemDim; k++) {
            u0(k,0) = nodes[k][min_idx];
          }

          for (Integer i = 0; i < 2; i++) { // iterate
            Matrix<Real> X_, dX_;
            for (Integer k = 0; k < ElemDim; k++) {
              u0(k,0) = std::min<Real>(1.0, u0(k,0));
              u0(k,0) = std::max<Real>(0.0, u0(k,0));
            }
            const auto eval_op = CoordBasis::SetupEval(Matrix<Real>(ElemDim,1,u0.begin(),false));
            CoordBasis::Eval(X_, Vector<CoordBasis>(CoordDim,(Iterator<CoordBasis>)X.begin()+src_idx*CoordDim,false),eval_op);
            CoordBasis::Eval(dX_, Vector<CoordBasis>(CoordDim*ElemDim,dX.begin()+src_idx*CoordDim*ElemDim,false),eval_op);

            const Tensor<Real,false,CoordDim,1> x0((Iterator<Real>)Xt_);
            const Tensor<Real,false,CoordDim,1> x(X_.begin());
            const Tensor<Real,false,CoordDim,ElemDim> x_u(dX_.begin());
            auto inv = [](const Tensor<Real,true,2,2>& M) {
              Tensor<Real,true,2,2> Minv;
              Real det_inv = 1.0 / (M(0,0)*M(1,1) - M(1,0)*M(0,1));
              Minv(0,0) = M(1,1) * det_inv;
              Minv(0,1) =-M(0,1) * det_inv;
              Minv(1,0) =-M(1,0) * det_inv;
              Minv(1,1) = M(0,0) * det_inv;
              return Minv;
            };
            auto du = inv(x_u.RotateRight()*x_u) * x_u.RotateRight()*(x0-x);
            u0 = u0 + du;

            auto x_u_squared = x_u.RotateRight() * x_u;
            adapt = sctl::sqrt<Real>( ((x0-x).RotateRight()*(x0-x))(0,0) / std::max<Real>(x_u_squared(0,0),x_u_squared(1,1)) );
          }
        }

        Matrix<Real> quad_nds;
        Vector<Real> quad_wts;
        DuffyQuad<ElemDim>(quad_nds, quad_wts, Vector<Real>(ElemDim,u0.begin(),false), order_singular, adapt);
        const CoordEvalOpType CoordEvalOp = CoordBasis::SetupEval(quad_nds);
        Integer Nnds = quad_wts.Dim();

        Vector<Real> X_, dX_, Xa_, Xn_;
        { // Set X_, dX_
          const Vector<CoordBasis> X__(CoordDim, (Iterator<CoordBasis>)X.begin() + src_idx * CoordDim, false);
          const Vector<CoordBasis> dX__(CoordDim * ElemDim, (Iterator<CoordBasis>)dX.begin() + src_idx * CoordDim * ElemDim, false);
          eval_basis(X_, X__, CoordDim, Nnds, CoordEvalOp);
          eval_basis(dX_, dX__, CoordDim * ElemDim, Nnds, CoordEvalOp);
        }
        if (CoordDim == 3 && ElemDim == 2) { // Compute Xa_, Xn_
          Xa_.ReInit(Nnds);
          Xn_.ReInit(Nnds*CoordDim);
          for (Long j = 0; j < Nnds; j++) {
            StaticArray<Real,CoordDim> normal;
            normal[0] = dX_[j*6+2]*dX_[j*6+5] - dX_[j*6+4]*dX_[j*6+3];
            normal[1] = dX_[j*6+4]*dX_[j*6+1] - dX_[j*6+0]*dX_[j*6+5];
            normal[2] = dX_[j*6+0]*dX_[j*6+3] - dX_[j*6+2]*dX_[j*6+1];
            Xa_[j] = sctl::sqrt<Real>(normal[0]*normal[0]+normal[1]*normal[1]+normal[2]*normal[2]);
            Real invXa = 1/Xa_[j];
            Xn_[j*3+0] = normal[0] * invXa;
            Xn_[j*3+1] = normal[1] * invXa;
            Xn_[j*3+2] = normal[2] * invXa;
          }
        }

        DensityEvalOpType DensityEvalOp;
        if (std::is_same<CoordBasis,DensityBasis>::value) {
          DensityEvalOp = CoordEvalOp;
        } else {
          DensityEvalOp = DensityBasis::SetupEval(quad_nds);
        }

        Matrix<Real> M__(Nnds * KDIM0, KDIM1);
        { // Set kernel matrix M__
          const Vector<Real> X0_(CoordDim, (Iterator<Real>)Xt.begin() + j * CoordDim, false);
          kernel.template KernelMatrix<Real>(M__, X0_, X_, Xn_);
        }
        for (Long k0 = 0; k0 < KDIM0; k0++) {
          for (Long k1 = 0; k1 < KDIM1; k1++) {
            for (Long l = 0; l < DensityBasis::Size(); l++) {
              Real M_lk = 0;
              for (Long n = 0; n < Nnds; n++) {
                Real quad_wt = Xa_[n] * quad_wts[n];
                M_lk += DensityEvalOp[l][n] * quad_wt * M__[n*KDIM0+k0][k1];
              }
              M[(j * KDIM0 + k0) * DensityBasis::Size() + l][k1] = M_lk;
            }
          }
        }
      }
      { // Set M (subtract direct)
        Matrix<Real> quad_nds;
        Vector<Real> quad_wts;
        TensorProductGaussQuad<ElemDim>(quad_nds, quad_wts, order_direct);
        const CoordEvalOpType CoordEvalOp = CoordBasis::SetupEval(quad_nds);
        Integer Nnds = quad_wts.Dim();

        Vector<Real> X_, dX_, Xa_, Xn_;
        { // Set X_, dX_
          eval_basis(X_, X, CoordDim, Nnds, CoordEvalOp);
          eval_basis(dX_, dX, CoordDim * ElemDim, Nnds, CoordEvalOp);
        }
        if (CoordDim == 3 && ElemDim == 2) { // Compute Xa_, Xn_
          Long N = Nelem*Nnds;
          Xa_.ReInit(N);
          Xn_.ReInit(N*CoordDim);
          for (Long j = 0; j < N; j++) {
            StaticArray<Real,CoordDim> normal;
            normal[0] = dX_[j*6+2]*dX_[j*6+5] - dX_[j*6+4]*dX_[j*6+3];
            normal[1] = dX_[j*6+4]*dX_[j*6+1] - dX_[j*6+0]*dX_[j*6+5];
            normal[2] = dX_[j*6+0]*dX_[j*6+3] - dX_[j*6+2]*dX_[j*6+1];
            Xa_[j] = sctl::sqrt<Real>(normal[0]*normal[0]+normal[1]*normal[1]+normal[2]*normal[2]);
            Real invXa = 1/Xa_[j];
            Xn_[j*3+0] = normal[0] * invXa;
            Xn_[j*3+1] = normal[1] * invXa;
            Xn_[j*3+2] = normal[2] * invXa;
          }
        }

        DensityEvalOpType DensityEvalOp;
        if (std::is_same<CoordBasis,DensityBasis>::value) {
          DensityEvalOp = CoordEvalOp;
        } else {
          DensityEvalOp = DensityBasis::SetupEval(quad_nds);
        }

        #pragma omp parallel for schedule(static)
        for (Long j = 0; j < Ninterac; j++) { // Subtract direct contribution
          const Long src_idx = pair_lst[j].first - elem_rank_offset;

          Matrix<Real> M__(Nnds * KDIM0, KDIM1);
          { // Set kernel matrix M__
            const Vector<Real> X0_(CoordDim, (Iterator<Real>)Xt.begin() + j * CoordDim, false);
            Vector<Real> X__(Nnds * CoordDim, X_.begin() + src_idx * Nnds * CoordDim, false);
            Vector<Real> Xn__(Nnds * CoordDim, Xn_.begin() + src_idx * Nnds * CoordDim, false);
            kernel.template KernelMatrix<Real>(M__, X0_, X__, Xn__);
          }
          for (Long k0 = 0; k0 < KDIM0; k0++) {
            for (Long k1 = 0; k1 < KDIM1; k1++) {
              for (Long l = 0; l < DensityBasis::Size(); l++) {
                Real M_lk = 0;
                for (Long n = 0; n < Nnds; n++) {
                  Real quad_wt = Xa_[src_idx * Nnds + n] * quad_wts[n];
                  M_lk += DensityEvalOp[l][n] * quad_wt * M__[n*KDIM0+k0][k1];
                }
                M[(j * KDIM0 + k0) * DensityBasis::Size() + l][k1] -= M_lk;
              }
            }
          }
        }
      }
    }

    template <class DensityBasis> static void EvalNearSingular(Vector<Real>& U, const Vector<DensityBasis>& density, const Matrix<Real>& M, const Vector<Pair<Long,Long>>& pair_lst, Long Nelem_, Long Ntrg_, Integer KDIM0_, Integer KDIM1_, const Comm& comm) {
      const Long Ninterac = pair_lst.Dim();
      const Integer dof = density.Dim() / Nelem_ / KDIM0_;
      SCTL_ASSERT(density.Dim() == Nelem_ * dof * KDIM0_);

      Long elem_rank_offset;
      { // Set elem_rank_offset
        comm.Scan(Ptr2ConstItr<Long>(&Nelem_,1), Ptr2Itr<Long>(&elem_rank_offset,1), 1, Comm::CommOp::SUM);
        elem_rank_offset -= Nelem_;
      }

      Vector<Real> U_loc(Ninterac*dof*KDIM1_);
      for (Long j = 0; j < Ninterac; j++) {
        const Long src_idx = pair_lst[j].first - elem_rank_offset;
        const Matrix<Real> M_(KDIM0_ * DensityBasis::Size(), KDIM1_, (Iterator<Real>)M[j * KDIM0_ * DensityBasis::Size()], false);
        Matrix<Real> U_(dof, KDIM1_, U_loc.begin() + j*dof*KDIM1_, false);
        Matrix<Real> F_(dof, KDIM0_ * DensityBasis::Size());
        for (Long i = 0; i < dof; i++) {
          for (Long k = 0; k < KDIM0_; k++) {
            for (Long l = 0; l < DensityBasis::Size(); l++) {
              F_[i][k * DensityBasis::Size() + l] = density[(src_idx * dof + i) * KDIM0_ + k][l];
            }
          }
        }
        Matrix<Real>::GEMM(U_, F_, M_);
      }

      if (U.Dim() != Ntrg_ * dof * KDIM1_) {
        U.ReInit(Ntrg_ * dof * KDIM1_);
        U = 0;
      }
      { // Set U
        Integer rank = comm.Rank();
        Integer np = comm.Size();

        Vector<Long> splitter_ranks;
        { // Set splitter_ranks
          Vector<Long> cnt(np);
          comm.Allgather(Ptr2ConstItr<Long>(&Ntrg_,1), 1, cnt.begin(), 1);
          scan(splitter_ranks, cnt);
        }

        Vector<Long> scatter_index, send_index, send_cnt(np), send_dsp(np);
        { // Set scatter_index, send_index, send_cnt, send_dsp
          { // Set scatter_index, send_index
            Vector<Pair<Long,Long>> scatter_pair(pair_lst.Dim());
            for (Long i = 0; i < pair_lst.Dim(); i++) {
              scatter_pair[i] = Pair<Long,Long>(pair_lst[i].second,i);
            }
            omp_par::merge_sort(scatter_pair.begin(), scatter_pair.end());

            send_index.ReInit(scatter_pair.Dim());
            scatter_index.ReInit(scatter_pair.Dim());
            for (Long i = 0; i < scatter_index.Dim(); i++) {
              send_index[i] = scatter_pair[i].first;
              scatter_index[i] = scatter_pair[i].second;
            }
          }
          for (Integer i = 0; i < np; i++) {
            send_dsp[i] = std::lower_bound(send_index.begin(), send_index.end(), splitter_ranks[i]) - send_index.begin();
          }
          for (Integer i = 0; i < np-1; i++) {
            send_cnt[i] = send_dsp[i+1] - send_dsp[i];
          }
          send_cnt[np-1] = send_index.Dim() - send_dsp[np-1];
        }

        Vector<Long> recv_index, recv_cnt(np), recv_dsp(np);
        { // Set recv_index, recv_cnt, recv_dsp
          comm.Alltoall(send_cnt.begin(), 1, recv_cnt.begin(), 1);
          scan(recv_dsp, recv_cnt);
          recv_index.ReInit(recv_cnt[np-1] + recv_dsp[np-1]);
          comm.Alltoallv(send_index.begin(), send_cnt.begin(), send_dsp.begin(), recv_index.begin(), recv_cnt.begin(), recv_dsp.begin());
        }

        Vector<Real> U_send(scatter_index.Dim() * dof * KDIM1_);
        for (Long i = 0; i < scatter_index.Dim(); i++) {
          Long idx = scatter_index[i]*dof*KDIM1_;
          for (Long k = 0; k < dof * KDIM1_; k++) {
            U_send[i*dof*KDIM1_ + k] = U_loc[idx + k];
          }
        }

        Vector<Real> U_recv(recv_index.Dim() * dof * KDIM1_);
        { // Set U_recv
          for (Long i = 0; i < np; i++) {
            send_cnt[i] *= dof * KDIM1_;
            send_dsp[i] *= dof * KDIM1_;
            recv_cnt[i] *= dof * KDIM1_;
            recv_dsp[i] *= dof * KDIM1_;
          }
          comm.Alltoallv(U_send.begin(), send_cnt.begin(), send_dsp.begin(), U_recv.begin(), recv_cnt.begin(), recv_dsp.begin());
        }

        for (Long i = 0; i < recv_index.Dim(); i++) { // Set U
          Long idx = (recv_index[i] - splitter_ranks[rank]) * dof * KDIM1_;
          for (Integer k = 0; k < dof * KDIM1_; k++) {
            U[idx + k] += U_recv[i*dof*KDIM1_ + k];
          }
        }
      }
    }



    template <class ElemList, class DensityBasis, class Kernel> static void Direct(Vector<Real>& U, const Vector<Real>& Xt, const ElemList& elem_lst, const Vector<DensityBasis>& density, const Kernel& kernel, Integer order_direct, const Comm& comm) {
      using CoordBasis = typename ElemList::CoordBasis;
      using CoordEvalOpType = typename CoordBasis::EvalOpType;
      using DensityEvalOpType = typename DensityBasis::EvalOpType;

      constexpr Integer CoordDim = ElemList::CoordDim();
      constexpr Integer ElemDim = ElemList::ElemDim();
      constexpr Integer KDIM0 = Kernel::SrcDim();
      constexpr Integer KDIM1 = Kernel::TrgDim();
      const Long Nelem = elem_lst.NElem();
      const Integer dof = density.Dim() / Nelem / KDIM0;
      SCTL_ASSERT(density.Dim() == Nelem * dof * KDIM0);

      Matrix<Real> quad_nds;
      Vector<Real> quad_wts;
      TensorProductGaussQuad<ElemDim>(quad_nds, quad_wts, order_direct);
      const CoordEvalOpType CoordEvalOp = CoordBasis::SetupEval(quad_nds);
      Integer Nnds = quad_wts.Dim();

      const Vector<CoordBasis>& X = elem_lst.ElemVector();
      Vector<CoordBasis> dX;
      CoordBasis::Grad(dX, X);

      Vector<Real> X_, dX_, Xa_, Xn_;
      eval_basis(X_, X, CoordDim, Nnds, CoordEvalOp);
      eval_basis(dX_, dX, CoordDim*ElemDim, Nnds, CoordEvalOp);
      if (CoordDim == 3 && ElemDim == 2) { // Compute Xa_, Xn_
        Long N = Nelem*Nnds;
        Xa_.ReInit(N);
        Xn_.ReInit(N*CoordDim);
        for (Long j = 0; j < N; j++) {
          StaticArray<Real,CoordDim> normal;
          normal[0] = dX_[j*6+2]*dX_[j*6+5] - dX_[j*6+4]*dX_[j*6+3];
          normal[1] = dX_[j*6+4]*dX_[j*6+1] - dX_[j*6+0]*dX_[j*6+5];
          normal[2] = dX_[j*6+0]*dX_[j*6+3] - dX_[j*6+2]*dX_[j*6+1];
          Xa_[j] = sctl::sqrt<Real>(normal[0]*normal[0]+normal[1]*normal[1]+normal[2]*normal[2]);
          Real invXa = 1/Xa_[j];
          Xn_[j*3+0] = normal[0] * invXa;
          Xn_[j*3+1] = normal[1] * invXa;
          Xn_[j*3+2] = normal[2] * invXa;
        }
      }

      Vector<Real> Fa_;
      { // Set Fa_
        Vector<Real> F_;
        if (std::is_same<CoordBasis,DensityBasis>::value) {
          eval_basis(F_, density, dof * KDIM0, Nnds, CoordEvalOp);
        } else {
          const DensityEvalOpType EvalOp = DensityBasis::SetupEval(quad_nds);
          eval_basis(F_, density, dof * KDIM0, Nnds, EvalOp);
        }

        Fa_.ReInit(F_.Dim());
        const Integer DensityDOF = dof * KDIM0;
        SCTL_ASSERT(F_.Dim() == Nelem * Nnds * DensityDOF);
        for (Long j = 0; j < Nelem; j++) {
          for (Integer k = 0; k < Nnds; k++) {
            Long idx = j * Nnds + k;
            Real quad_wt = Xa_[idx] * quad_wts[k];
            for (Integer l = 0; l < DensityDOF; l++) {
              Fa_[idx * DensityDOF + l] = F_[idx * DensityDOF + l] * quad_wt;
            }
          }
        }
      }

      { // Evaluate potential
        const Long Ntrg = Xt.Dim() / CoordDim;
        SCTL_ASSERT(Xt.Dim() == Ntrg * CoordDim);
        if (U.Dim() != Ntrg * dof * KDIM1) {
          U.ReInit(Ntrg * dof * KDIM1);
          U = 0;
        }
        ParticleFMM<Real,CoordDim>::Eval(U, Xt, X_, Xn_, Fa_, kernel, comm);
      }
    }

  public:

    template <class DensityBasis, class ElemList, class Kernel> void Setup(const ElemList& elem_lst, const Vector<Real>& Xt, const Kernel& kernel, Integer order_singular, Integer order_direct, Real period_length, const Comm& comm) {
      Xt_.ReInit(0);
      M_singular.ReInit(0,0);
      M_near_singular.ReInit(0,0);
      pair_lst.ReInit(0);

      order_direct_ = order_direct;
      period_length_ = period_length;
      comm_ = comm;

      Profile::Tic("Setup", &comm_);
      static_assert(std::is_same<Real,typename DensityBasis::ValueType>::value);
      static_assert(std::is_same<Real,typename ElemList::CoordType>::value);
      static_assert(DensityBasis::Dim() == ElemList::ElemDim());

      Xt_ = Xt;
      M_singular.ReInit(0,0);

      Profile::Tic("SetupNearSingular", &comm_);
      SetupNearSingular<DensityBasis>(M_near_singular, pair_lst, Xt_, Vector<Long>(), elem_lst, kernel, order_singular, order_direct_, period_length_, comm_);
      Profile::Toc();

      Profile::Toc();
    }

    template <class DensityBasis, class PotentialBasis, class ElemList, class Kernel> void Setup(const ElemList& elem_lst, const Kernel& kernel, Integer order_singular, Integer order_direct, Real period_length, const Comm& comm, Real Rqbx = 0) {
      Xt_.ReInit(0);
      M_singular.ReInit(0,0);
      M_near_singular.ReInit(0,0);
      pair_lst.ReInit(0);

      order_direct_ = order_direct;
      period_length_ = period_length;
      comm_ = comm;

      Profile::Tic("Setup", &comm_);
      static_assert(std::is_same<Real,typename PotentialBasis::ValueType>::value);
      static_assert(std::is_same<Real,typename DensityBasis::ValueType>::value);
      static_assert(std::is_same<Real,typename ElemList::CoordType>::value);
      static_assert(PotentialBasis::Dim() == ElemList::ElemDim());
      static_assert(DensityBasis::Dim() == ElemList::ElemDim());

      Vector<Long> trg_surf;
      { // Set Xt_
        using CoordBasis = typename ElemList::CoordBasis;
        Matrix<Real> trg_nds = PotentialBasis::Nodes();
        auto Meval = CoordBasis::SetupEval(trg_nds);
        eval_basis(Xt_, elem_lst.ElemVector(), ElemList::CoordDim(), trg_nds.Dim(1), Meval);

        { // Set trg_surf
          const Long Nelem = elem_lst.NElem();
          const Long Nnds  = trg_nds.Dim(1);
          Long elem_offset;
          { // Set elem_offset
            comm.Scan(Ptr2ConstItr<Long>(&Nelem,1), Ptr2Itr<Long>(&elem_offset,1), 1, Comm::CommOp::SUM);
            elem_offset -= Nelem;
          }
          trg_surf.ReInit(elem_lst.NElem() * trg_nds.Dim(1));
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnds; j++) {
              trg_surf[i*Nnds+j] = elem_offset + i;
            }
          }
        }
      }

      Profile::Tic("SetupSingular", &comm_);
      SetupSingular<DensityBasis>(M_singular, PotentialBasis::Nodes(), elem_lst, kernel, order_singular, order_direct_, Rqbx);
      Profile::Toc();

      Profile::Tic("SetupNearSingular", &comm_);
      SetupNearSingular<DensityBasis>(M_near_singular, pair_lst, Xt_, trg_surf, elem_lst, kernel, order_singular, order_direct_, period_length_, comm_);
      Profile::Toc();

      Profile::Toc();
    }

    template <class DensityBasis, class PotentialBasis, class ElemList, class Kernel> void Eval(Vector<PotentialBasis>& U, const ElemList& elements, const Vector<DensityBasis>& F, const Kernel& kernel) {
      Profile::Tic("Eval", &comm_);
      Matrix<Real> U_singular;
      Vector<Real> U_direct, U_near_sing;

      Profile::Tic("EvalDirect", &comm_);
      Direct(U_direct, Xt_, elements, F, kernel, order_direct_, comm_);
      Profile::Toc();

      Profile::Tic("EvalSingular", &comm_);
      EvalSingular(U_singular, F, M_singular, kernel.SrcDim(), kernel.TrgDim());
      Profile::Toc();

      Profile::Tic("EvalNearSingular", &comm_);
      EvalNearSingular(U_near_sing, F, M_near_singular, pair_lst, elements.NElem(), Xt_.Dim() / ElemList::CoordDim(), kernel.SrcDim(), kernel.TrgDim(), comm_);
      SCTL_ASSERT(U_near_sing.Dim() == U_direct.Dim());
      Profile::Toc();

      const Long dof = U_direct.Dim() / (elements.NElem() * PotentialBasis::Size() * kernel.TrgDim());
      SCTL_ASSERT(U_direct   .Dim() == elements.NElem() * PotentialBasis::Size() * dof * kernel.TrgDim());
      SCTL_ASSERT(U_near_sing.Dim() == elements.NElem() * PotentialBasis::Size() * dof * kernel.TrgDim());

      if (U.Dim() != elements.NElem() * dof * kernel.TrgDim()) {
        U.ReInit(elements.NElem() * dof * kernel.TrgDim());
      }
      for (int i = 0; i < elements.NElem(); i++) {
        for (int j = 0; j < PotentialBasis::Size(); j++) {
          for (int k = 0; k < dof*kernel.TrgDim(); k++) {
            Real& U_ = U[i*dof*kernel.TrgDim()+k][j];
            U_ = 0;
            U_ += U_direct   [(i*PotentialBasis::Size()+j)*dof*kernel.TrgDim()+k];
            U_ += U_near_sing[(i*PotentialBasis::Size()+j)*dof*kernel.TrgDim()+k];
            U_ *= kernel.template ScaleFactor<Real>();
          }
        }
      }
      if (U_singular.Dim(1)) {
        SCTL_ASSERT(U_singular.Dim(0) == elements.NElem() * dof * kernel.TrgDim());
        SCTL_ASSERT(U_singular.Dim(1) == PotentialBasis::Size());
        for (int i = 0; i < elements.NElem(); i++) {
          for (int j = 0; j < PotentialBasis::Size(); j++) {
            for (int k = 0; k < dof*kernel.TrgDim(); k++) {
              U[i*dof*kernel.TrgDim()+k][j] += U_singular[i*dof*kernel.TrgDim()+k][j] * kernel.template ScaleFactor<Real>();
            }
          }
        }
      }
      Profile::Toc();
    }

    template <class DensityBasis, class ElemList, class Kernel> void Eval(Vector<Real>& U, const ElemList& elements, const Vector<DensityBasis>& F, const Kernel& kernel) {
      Profile::Tic("Eval", &comm_);
      Matrix<Real> U_singular;
      Vector<Real> U_direct, U_near_sing;

      Profile::Tic("EvalDirect", &comm_);
      Direct(U_direct, Xt_, elements, F, kernel, order_direct_, comm_);
      Profile::Toc();

      Profile::Tic("EvalSingular", &comm_);
      EvalSingular(U_singular, F, M_singular, kernel.SrcDim(), kernel.TrgDim());
      Profile::Toc();

      Profile::Tic("EvalNearSingular", &comm_);
      EvalNearSingular(U_near_sing, F, M_near_singular, pair_lst, elements.NElem(), Xt_.Dim() / ElemList::CoordDim(), kernel.SrcDim(), kernel.TrgDim(), comm_);
      SCTL_ASSERT(U_near_sing.Dim() == U_direct.Dim());
      Profile::Toc();


      Long Nt = Xt_.Dim() / ElemList::CoordDim();
      const Long dof = U_direct.Dim() / (Nt * kernel.TrgDim());
      SCTL_ASSERT(U_direct.Dim() == Nt * dof * kernel.TrgDim());

      if (U.Dim() != U_direct.Dim()) {
        U.ReInit(U_direct.Dim());
      }
      for (int i = 0; i < U.Dim(); i++) {
        U[i] = (U_direct[i] + U_near_sing[i]) * kernel.template ScaleFactor<Real>();
      }
      if (U_singular.Dim(1)) {
        SCTL_ASSERT(U_singular.Dim(0) == elements.NElem() * dof * kernel.TrgDim());
        const Long Nnodes = U_singular.Dim(1);
        for (int i = 0; i < elements.NElem(); i++) {
          for (int j = 0; j < Nnodes; j++) {
            for (int k = 0; k < dof*kernel.TrgDim(); k++) {
              Real& U_ = U[(i*Nnodes+j)*dof*kernel.TrgDim()+k];
              U_ += U_singular[i*dof*kernel.TrgDim()+k][j] * kernel.template ScaleFactor<Real>();
            }
          }
        }
      }
      Profile::Toc();
    }

    template <Integer ORDER = 5> static void test(Integer order_singular = 10, Integer order_direct = 5, const Comm& comm = Comm::World()) {
      constexpr Integer COORD_DIM = 3;
      constexpr Integer ELEM_DIM = COORD_DIM-1;
      using ElemList = ElemList<COORD_DIM, Basis<Real, ELEM_DIM, ORDER>>;
      using DensityBasis = Basis<Real, ELEM_DIM, ORDER>;
      using PotentialBasis = Basis<Real, ELEM_DIM, ORDER>;

      int np = comm.Size();
      int rank = comm.Rank();
      auto build_torus = [rank,np](ElemList& elements, long Nt, long Np, Real Rmajor, Real Rminor){
        auto nodes = ElemList::CoordBasis::Nodes();
        auto torus = [](Real theta, Real phi, Real Rmajor, Real Rminor) {
          Real R = Rmajor + Rminor * cos<Real>(phi);
          Real X = R * cos<Real>(theta);
          Real Y = R * sin<Real>(theta);
          Real Z = Rminor * sin<Real>(phi);
          return std::make_tuple(X,Y,Z);
        };

        long start = Nt*Np*(rank+0)/np;
        long end   = Nt*Np*(rank+1)/np;
        elements.ReInit(end - start);
        for (long ii = start; ii < end; ii++) {
          long i = ii / Np;
          long j = ii % Np;
          for (int k = 0; k < ElemList::CoordBasis::Size(); k++) {
            Real X, Y, Z;
            Real theta = 2 * const_pi<Real>() * (i + nodes[0][k]) / Nt;
            Real phi   = 2 * const_pi<Real>() * (j + nodes[1][k]) / Np;
            std::tie(X,Y,Z) = torus(theta, phi, Rmajor, Rminor);
            elements(ii-start,0)[k] = X;
            elements(ii-start,1)[k] = Y;
            elements(ii-start,2)[k] = Z;
          }
        }
      };
      ElemList elements_src, elements_trg;
      build_torus(elements_src, 28, 16, 2, 1.0);
      build_torus(elements_trg, 29, 17, 2, 0.99);

      Vector<Real> Xt;
      Vector<PotentialBasis> U_onsurf, U_offsurf;
      Vector<DensityBasis> density_sl, density_dl;
      { // Set Xt, elements_src, elements_trg, density_sl, density_dl, U
        Real X0[COORD_DIM] = {3,2,1};
        std::function<void(Real*,Real*,Real*)> potential = [X0](Real* U, Real* X, Real* Xn) {
          Real dX[COORD_DIM] = {X[0]-X0[0],X[1]-X0[1],X[2]-X0[2]};
          Real Rinv = 1/sqrt(dX[0]*dX[0]+dX[1]*dX[1]+dX[2]*dX[2]);
          U[0] = Rinv;
        };
        std::function<void(Real*,Real*,Real*)> potential_normal_derivative = [X0](Real* U, Real* X, Real* Xn) {
          Real dX[COORD_DIM] = {X[0]-X0[0],X[1]-X0[1],X[2]-X0[2]};
          Real Rinv = 1/sqrt(dX[0]*dX[0]+dX[1]*dX[1]+dX[2]*dX[2]);
          Real RdotN = dX[0]*Xn[0]+dX[1]*Xn[1]+dX[2]*Xn[2];
          U[0] = -RdotN * Rinv*Rinv*Rinv;
        };

        DiscretizeSurfaceFn<COORD_DIM,1>(density_sl, elements_src, potential_normal_derivative);
        DiscretizeSurfaceFn<COORD_DIM,1>(density_dl, elements_src, potential);
        DiscretizeSurfaceFn<COORD_DIM,1>(U_onsurf  , elements_src, potential);
        DiscretizeSurfaceFn<COORD_DIM,1>(U_offsurf , elements_trg, potential);

        for (long i = 0; i < elements_trg.NElem(); i++) { // Set Xt
          for (long j = 0; j < PotentialBasis::Size(); j++) {
            for (int k = 0; k < COORD_DIM; k++) {
              Xt.PushBack(elements_trg(i,k)[j]);
            }
          }
        }
      }

      GenericKernel<Laplace3D_DxU> Laplace_DxU;
      GenericKernel<Laplace3D_FxU> Laplace_FxU;

      Profile::Enable(true);
      if (1) { // Greeen's identity test (Laplace, on-surface)
        Profile::Tic("OnSurface", &comm);
        Quadrature<Real> quadrature_DxU, quadrature_FxU;
        quadrature_FxU.Setup<DensityBasis, PotentialBasis>(elements_src, Laplace_FxU, order_singular, order_direct, -1.0, comm);
        quadrature_DxU.Setup<DensityBasis, PotentialBasis>(elements_src, Laplace_DxU, order_singular, order_direct, -1.0, comm);

        Vector<PotentialBasis> U_sl, U_dl;
        quadrature_FxU.Eval(U_sl, elements_src, density_sl, Laplace_FxU);
        quadrature_DxU.Eval(U_dl, elements_src, density_dl, Laplace_DxU);
        Profile::Toc();

        Real max_err = 0;
        Vector<PotentialBasis> err(U_onsurf.Dim());
        for (long i = 0; i < U_sl.Dim(); i++) {
          for (long j = 0; j < PotentialBasis::Size(); j++) {
            err[i][j] = 0.5*U_onsurf[i][j] - (U_sl[i][j] + U_dl[i][j]);
            max_err = std::max<Real>(max_err, fabs(err[i][j]));
          }
        }
        { // Print error
          Real glb_err;
          comm.Allreduce(Ptr2ConstItr<Real>(&max_err,1), Ptr2Itr<Real>(&glb_err,1), 1, Comm::CommOp::MAX);
          if (!comm.Rank()) std::cout<<"Error = "<<glb_err<<'\n';
        }
        { // Write VTK output
          VTUData vtu;
          vtu.AddElems(elements_src, err, ORDER);
          vtu.WriteVTK("err", comm);
        }
        { // Write VTK output
          VTUData vtu;
          vtu.AddElems(elements_src, U_onsurf, ORDER);
          vtu.WriteVTK("U", comm);
        }
      }
      if (1) { // Greeen's identity test (Laplace, off-surface)
        Profile::Tic("OffSurface", &comm);
        Quadrature<Real> quadrature_DxU, quadrature_FxU;
        quadrature_FxU.Setup<DensityBasis>(elements_src, Xt, Laplace_FxU, order_singular, order_direct, -1.0, comm);
        quadrature_DxU.Setup<DensityBasis>(elements_src, Xt, Laplace_DxU, order_singular, order_direct, -1.0, comm);

        Vector<Real> U_sl, U_dl;
        quadrature_FxU.Eval(U_sl, elements_src, density_sl, Laplace_FxU);
        quadrature_DxU.Eval(U_dl, elements_src, density_dl, Laplace_DxU);
        Profile::Toc();

        Real max_err = 0;
        Vector<PotentialBasis> err(elements_trg.NElem());
        for (long i = 0; i < elements_trg.NElem(); i++) {
          for (long j = 0; j < PotentialBasis::Size(); j++) {
            err[i][j] = U_offsurf[i][j] - (U_sl[i*PotentialBasis::Size()+j] + U_dl[i*PotentialBasis::Size()+j]);
            max_err = std::max<Real>(max_err, fabs(err[i][j]));
          }
        }
        { // Print error
          Real glb_err;
          comm.Allreduce(Ptr2ConstItr<Real>(&max_err,1), Ptr2Itr<Real>(&glb_err,1), 1, Comm::CommOp::MAX);
          if (!comm.Rank()) std::cout<<"Error = "<<glb_err<<'\n';
        }
        { // Write VTK output
          VTUData vtu;
          vtu.AddElems(elements_trg, err, ORDER);
          vtu.WriteVTK("err", comm);
        }
        { // Write VTK output
          VTUData vtu;
          vtu.AddElems(elements_trg, U_offsurf, ORDER);
          vtu.WriteVTK("U", comm);
        }
      }
      Profile::print(&comm);
    }

    static void test1() {
      const Comm& comm = Comm::World();
      constexpr Integer ORDER = 15;
      Integer order_singular = 20;
      Integer order_direct = 20;

      constexpr Integer COORD_DIM = 3;
      constexpr Integer ELEM_DIM = COORD_DIM-1;
      using ElemList = ElemList<COORD_DIM, Basis<Real, ELEM_DIM, ORDER>>;
      using DensityBasis = Basis<Real, ELEM_DIM, ORDER>;
      using PotentialBasis = Basis<Real, ELEM_DIM, ORDER>;

      int np = comm.Size();
      int rank = comm.Rank();
      auto build_sphere = [rank,np](ElemList& elements, Real X, Real Y, Real Z, Real R){
        auto nodes = ElemList::CoordBasis::Nodes();

        long start = 2*COORD_DIM*(rank+0)/np;
        long end   = 2*COORD_DIM*(rank+1)/np;
        elements.ReInit(end - start);
        for (long ii = start; ii < end; ii++) {
          long i = ii / 2;
          long j = ii % 2;
          for (int k = 0; k < ElemList::CoordBasis::Size(); k++) {
            Real coord[COORD_DIM];
            coord[(i+0)%COORD_DIM] = (j ? -1.0 : 1.0);
            coord[(i+1)%COORD_DIM] = 2.0 * nodes[j?1:0][k] - 1.0;
            coord[(i+2)%COORD_DIM] = 2.0 * nodes[j?0:1][k] - 1.0;
            Real R0 = sqrt<Real>(coord[0]*coord[0] + coord[1]*coord[1] + coord[2]*coord[2]);

            elements(ii-start,0)[k] = X + R * coord[0] / R0;
            elements(ii-start,1)[k] = Y + R * coord[1] / R0;
            elements(ii-start,2)[k] = Z + R * coord[2] / R0;
          }
        }
      };

      ElemList elements;
      build_sphere(elements, 0.0, 0.0, 0.0, 1.00);

      Vector<DensityBasis> density_sl;
      { // Set density_sl
        std::function<void(Real*,Real*,Real*)> sigma = [](Real* U, Real* X, Real* Xn) {
          Real R = sqrt(X[0]*X[0]+X[1]*X[1]+X[2]*X[2]);
          Real sinp = sqrt(X[1]*X[1] + X[2]*X[2]) / R;
          Real cosp = -X[0] / R;

          U[0] = -1.5;
          U[1] = 0;
          U[2] = 0;
        };
        DiscretizeSurfaceFn<COORD_DIM,3>(density_sl, elements, sigma);
      }

      GenericKernel<Stokes3D_DxU> Stokes_DxU;
      GenericKernel<Stokes3D_FxU> Stokes_FxU;

      Profile::Enable(true);
      if (1) {
        Vector<PotentialBasis> U;
        Quadrature<Real> quadrature_FxU;
        quadrature_FxU.Setup<DensityBasis, PotentialBasis>(elements, Stokes_FxU, order_singular, order_direct, -1.0, comm);
        quadrature_FxU.Eval(U, elements, density_sl, Stokes_FxU);

        { // Write VTK output
          VTUData vtu;
          vtu.AddElems(elements, U, ORDER);
          vtu.WriteVTK("U", comm);
        }
        { // Write VTK output
          VTUData vtu;
          vtu.AddElems(elements, density_sl, ORDER);
          vtu.WriteVTK("sigma", comm);
        }
      }
      Profile::print(&comm);
    }

  private:

    static void scan(Vector<Long>& dsp, const Vector<Long>& cnt) {
      dsp.ReInit(cnt.Dim());
      if (cnt.Dim()) dsp[0] = 0;
      omp_par::scan(cnt.begin(), dsp.begin(), cnt.Dim());
    }

    template <class Basis> static void eval_basis(Vector<Real>& value, const Vector<Basis> X, Integer dof, Integer Nnds, const typename Basis::EvalOpType& EvalOp) {
      Long Nelem = X.Dim() / dof;
      SCTL_ASSERT(X.Dim() == Nelem * dof);

      value.ReInit(Nelem*Nnds*dof);
      Matrix<Real> X_(Nelem*dof, Nnds, value.begin(),false);
      Basis::Eval(X_, X, EvalOp);
      for (Long j = 0; j < Nelem; j++) { // Rearrange data
        Matrix<Real> X(Nnds, dof, X_[j*dof], false);
        X = Matrix<Real>(dof, Nnds, X_[j*dof], false).Transpose();
      }
    }

    template <int CoordDim, int FnDim, class FnBasis, class ElemList> static void DiscretizeSurfaceFn(Vector<FnBasis>& U, const ElemList& elements, std::function<void(Real*,Real*,Real*)> fn) {
      using CoordBasis = typename ElemList::CoordBasis;
      const long Nelem = elements.NElem();
      U.ReInit(Nelem * FnDim);

      Matrix<Real> X, X_grad;
      { // Set X, X_grad
        Vector<CoordBasis> coord = elements.ElemVector();
        Vector<CoordBasis> coord_grad;
        CoordBasis::Grad(coord_grad, coord);

        const auto Meval = CoordBasis::SetupEval(FnBasis::Nodes());
        CoordBasis::Eval(X, coord, Meval);
        CoordBasis::Eval(X_grad, coord_grad, Meval);
      }

      for (long i = 0; i < Nelem; i++) {
        for (long j = 0; j < FnBasis::Size(); j++) {
          Real X_[CoordDim], Xn[CoordDim], U_[FnDim];
          for (long k = 0; k < CoordDim; k++) {
            X_[k] = X[i*CoordDim+k][j];
          }
          { // Set Xn
            Real Xu[CoordDim], Xv[CoordDim];
            for (long k = 0; k < CoordDim; k++) {
              Xu[k] = X_grad[(i*CoordDim+k)*2+0][j];
              Xv[k] = X_grad[(i*CoordDim+k)*2+1][j];
            }
            Real dA = 0;
            for (long k = 0; k < CoordDim; k++) {
              Xn[k] = Xu[(k+1)%CoordDim] * Xv[(k+2)%CoordDim];
              Xn[k] -= Xv[(k+1)%CoordDim] * Xu[(k+2)%CoordDim];
              dA += Xn[k] * Xn[k];
            }
            dA = sqrt(dA);
            for (long k = 0; k < CoordDim; k++) {
              Xn[k] /= dA;
            }
          }
          fn(U_, X_, Xn);
          for (long k = 0; k < FnDim; k++) {
            U[i*FnDim+k][j] = U_[k];
          }
        }
      }
    }

    Vector<Real> Xt_;
    Matrix<Real> M_singular;
    Matrix<Real> M_near_singular;
    Vector<Pair<Long,Long>> pair_lst;
    Integer order_direct_;

    Real period_length_;
    Comm comm_;
};


template <class Real, Integer ORDER=10> class Stellarator {
  private:
    static constexpr Integer COORD_DIM = 3;
    static constexpr Integer ELEM_DIM = COORD_DIM-1;
    using ElemBasis = Basis<Real, ELEM_DIM, ORDER>;
    using ElemLst = ElemList<COORD_DIM, ElemBasis>;

    struct Laplace3D_dUxF {
      template <class ValueType> static constexpr ValueType ScaleFactor() {
        return 1 / (4 * const_pi<ValueType>());
      }
      template <class ValueType> static void Eval(ValueType (&u)[3][1], const ValueType (&r)[3], const ValueType (&n)[3], void* ctx_ptr) {
        ValueType r2 = r[0]*r[0]+r[1]*r[1]+r[2]*r[2];
        ValueType rinv = (r2>1e-16 ? 1/sqrt<ValueType>(r2) : 0);
        ValueType rinv3 = rinv * rinv * rinv;
        u[0][0] = -r[0] * rinv3;
        u[1][0] = -r[1] * rinv3;
        u[2][0] = -r[2] * rinv3;
      }
    };

    struct BiotSavart3D {
      template <class ValueType> static constexpr ValueType ScaleFactor() {
        return 1 / (4 * const_pi<ValueType>());
      }
      template <class ValueType> static void Eval(ValueType (&u)[3][3], const ValueType (&r)[3], const ValueType (&n)[3], void* ctx_ptr) {
        ValueType r2 = r[0]*r[0]+r[1]*r[1]+r[2]*r[2];
        ValueType rinv = (r2>1e-16 ? 1/sqrt<ValueType>(r2) : 0);
        ValueType rinv3 = rinv * rinv * rinv;
        u[0][0] =   (0) * rinv3; u[0][1] = -r[2] * rinv3; u[0][2] =  r[1] * rinv3;
        u[1][0] =  r[2] * rinv3; u[1][1] =   (0) * rinv3; u[1][2] = -r[0] * rinv3;
        u[2][0] = -r[1] * rinv3; u[2][1] =  r[0] * rinv3; u[2][2] =   (0) * rinv3;
      }
    };

    struct Laplace3D_dUxD {
      template <class ValueType> static constexpr ValueType ScaleFactor() {
        return 1 / (4 * const_pi<ValueType>());
      }
      template <class ValueType> static void Eval(ValueType (&u)[3][3], const ValueType (&r)[3], const ValueType (&n)[3], void* ctx_ptr) {
        ValueType r2 = r[0]*r[0]+r[1]*r[1]+r[2]*r[2];
        ValueType rinv = (r2>1e-16 ? 1/sqrt<ValueType>(r2) : 0);
        ValueType rdotn = r[0]*n[0] + r[1]*n[1] + r[2]*n[2];
        ValueType rinv2 = rinv * rinv;
        ValueType rinv3 = rinv * rinv2;
        ValueType rinv5 = rinv3 * rinv2;

        u[0][0] = -1 * rinv3 + 3 * r[0] * r[0] * rinv5;
        u[0][1] = -0 * rinv3 + 3 * r[0] * r[1] * rinv5;
        u[0][2] = -0 * rinv3 + 3 * r[0] * r[2] * rinv5;

        u[1][0] = -0 * rinv3 + 3 * r[1] * r[0] * rinv5;
        u[1][1] = -1 * rinv3 + 3 * r[1] * r[1] * rinv5;
        u[1][2] = -0 * rinv3 + 3 * r[1] * r[2] * rinv5;

        u[2][0] = -0 * rinv3 + 3 * r[2] * r[0] * rinv5;
        u[2][1] = -0 * rinv3 + 3 * r[2] * r[1] * rinv5;
        u[2][2] = -1 * rinv3 + 3 * r[2] * r[2] * rinv5;
      }
    };

    struct Laplace3D_DxdU {
      template <class ValueType> static constexpr ValueType ScaleFactor() {
        return 1 / (4 * const_pi<ValueType>());
      }
      template <class ValueType> static void Eval(ValueType (&u)[1][3], const ValueType (&r)[3], const ValueType (&n)[3], void* ctx_ptr) {
        ValueType r2 = r[0]*r[0]+r[1]*r[1]+r[2]*r[2];
        ValueType rinv = (r2>1e-16 ? 1/sqrt<ValueType>(r2) : 0);
        ValueType rdotn = r[0]*n[0] + r[1]*n[1] + r[2]*n[2];
        ValueType rinv2 = rinv * rinv;
        ValueType rinv3 = rinv * rinv2;
        ValueType rinv5 = rinv3 * rinv2;
        u[0][0] = -n[0] * rinv3 + 3*rdotn * r[0] * rinv5;
        u[0][1] = -n[1] * rinv3 + 3*rdotn * r[1] * rinv5;
        u[0][2] = -n[2] * rinv3 + 3*rdotn * r[2] * rinv5;
      }
    };

    struct Laplace3D_Fxd2U {
      template <class ValueType> static constexpr ValueType ScaleFactor() {
        return 1 / (4 * const_pi<ValueType>());
      }
      template <class ValueType> static void Eval(ValueType (&u)[1][9], const ValueType (&r)[3], const ValueType (&n)[3], void* ctx_ptr) {
        ValueType r2 = r[0]*r[0]+r[1]*r[1]+r[2]*r[2];
        ValueType rinv = (r2>1e-16 ? 1/sqrt<ValueType>(r2) : 0);
        ValueType rinv2 = rinv * rinv;
        ValueType rinv3 = rinv * rinv2;
        ValueType rinv5 = rinv3 * rinv2;

        u[0][0+3*0] = -1 * rinv3 + 3 * r[0] * r[0] * rinv5;
        u[0][1+3*0] = -0 * rinv3 + 3 * r[0] * r[1] * rinv5;
        u[0][2+3*0] = -0 * rinv3 + 3 * r[0] * r[2] * rinv5;

        u[0][0+3*1] = -0 * rinv3 + 3 * r[1] * r[0] * rinv5;
        u[0][1+3*1] = -1 * rinv3 + 3 * r[1] * r[1] * rinv5;
        u[0][2+3*1] = -0 * rinv3 + 3 * r[1] * r[2] * rinv5;

        u[0][0+3*2] = -0 * rinv3 + 3 * r[2] * r[0] * rinv5;
        u[0][1+3*2] = -0 * rinv3 + 3 * r[2] * r[1] * rinv5;
        u[0][2+3*2] = -1 * rinv3 + 3 * r[2] * r[2] * rinv5;
      }
    };

  public:
    Stellarator(const Vector<Long>& NtNp = Vector<Long>()) {
      NtNp_ = NtNp;
      Long Nsurf = NtNp_.Dim() / 2;
      SCTL_ASSERT(Nsurf*2 == NtNp_.Dim());

      Long Nelem = 0;
      elem_dsp.ReInit(Nsurf);
      if (elem_dsp.Dim()) elem_dsp[0] = 0;
      for (Long i = 0; i < Nsurf; i++) {
        Nelem += NtNp_[i*2+0]*NtNp_[i*2+1];
        if (i+1 < Nsurf) elem_dsp[i+1] = elem_dsp[i] + NtNp_[i*2+0]*NtNp_[i*2+1];
      }
      elements.ReInit(Nelem);
      for (Long i = 0; i < Nsurf; i++) {
        InitSurf(i);
      }
    }

    Long ElemIdx(Long s, Long t, Long p) {
      SCTL_ASSERT(0 <= s && s < elem_dsp.Dim());
      SCTL_ASSERT(0 <= t && t < NtNp_[s*2+0]);
      SCTL_ASSERT(0 <= p && p < NtNp_[s*2+1]);
      return elem_dsp[s] + t*NtNp_[s*2+1] + p;
    }
    ElemBasis& Elem(Long elem, Integer dim) {
      return elements(elem,dim);
    }
    const ElemBasis& Elem(Long elem, Integer dim) const {
      return elements(elem,dim);
    }
    const ElemLst& GetElemList() {
      return elements;
    }

    static void test_() {
      constexpr Integer order_singular = 20;
      constexpr Integer order_direct = 35;
      Comm comm = Comm::World();
      Profile::Enable(true);

      Stellarator<Real,ORDER> S;
      { // Set S
        Vector<Real> X(COORD_DIM);
        Vector<Real> R(1);
        X = 0;
        R = 1;
        SCTL_ASSERT(X.Dim() == R.Dim() * COORD_DIM);
        Long N = R.Dim();
        S.elements.ReInit(2*COORD_DIM*N);
        auto nodes = ElemLst::CoordBasis::Nodes();
        for (Long l = 0; l < N; l++) {
          for (Integer i = 0; i < COORD_DIM; i++) {
            for (Integer j = 0; j < 2; j++) {
              for (int k = 0; k < ElemLst::CoordBasis::Size(); k++) {
                Real coord[COORD_DIM];
                coord[(i+0)%COORD_DIM] = (j ? -1.0 : 1.0);
                coord[(i+1)%COORD_DIM] = 2.0 * nodes[j?1:0][k] - 1.0;
                coord[(i+2)%COORD_DIM] = 2.0 * nodes[j?0:1][k] - 1.0;
                Real R0 = sqrt<Real>(coord[0]*coord[0] + coord[1]*coord[1] + coord[2]*coord[2]);

                S.elements((l*COORD_DIM+i)*2+j,0)[k] = X[l*COORD_DIM+0] + R[l] * coord[0] / R0;
                S.elements((l*COORD_DIM+i)*2+j,1)[k] = X[l*COORD_DIM+1] + R[l] * coord[1] / R0;
                S.elements((l*COORD_DIM+i)*2+j,2)[k] = X[l*COORD_DIM+2] + R[l] * coord[2] / R0;
              }
            }
          }
        }
        S.elem_dsp.ReInit(1);
        S.elem_dsp = 0;
      }

      S.quadrature_Fxd2U.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_Fxd2U, order_singular, order_direct, -1.0, comm, -0.01 * pow<-2,Real>(ORDER));
      //S.quadrature_FxdU.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxdU, order_singular, order_direct, -1.0, comm, -0.01 * pow<-2,Real>(ORDER));

      { // test Fxd2U
        Vector<ElemBasis> U, sigma(S.elements.NElem());
        sigma = 1;
        sigma[0] = 1;
        S.quadrature_Fxd2U.Eval(U, S.GetElemList(), sigma, S.Laplace_Fxd2U);
        //S.quadrature_FxdU.Eval(U, S.GetElemList(), sigma, S.Laplace_FxdU);
        { // Write VTU
          VTUData vtu;
          vtu.AddElems(S.GetElemList(), U, ORDER);
          vtu.WriteVTK("test", comm);
        }
      }

      Profile::print(&comm);
    }

    static void test() {
      constexpr Integer order_singular = 15;
      constexpr Integer order_direct = 35;
      Comm comm = Comm::World();
      Profile::Enable(true);

      Stellarator<Real,ORDER> S;
      { // Init S
        Vector<Long> NtNp;
        NtNp.PushBack(40);
        NtNp.PushBack(8);
        S = Stellarator<Real,ORDER>(NtNp);
      }

      Vector<ElemBasis> normal, area_elem;
      auto compute_dot_prod = [](const Vector<ElemBasis>& A, const Vector<ElemBasis>& B) {
        const Long Nelem = A.Dim() / COORD_DIM;
        const Long Nnodes = ElemBasis::Size();
        SCTL_ASSERT(A.Dim() == Nelem * COORD_DIM);
        SCTL_ASSERT(B.Dim() == Nelem * COORD_DIM);
        Vector<ElemBasis> AdotB(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            Real a_dot_b = 0;
            a_dot_b += A[i*COORD_DIM+0][j]*B[i*COORD_DIM+0][j];
            a_dot_b += A[i*COORD_DIM+1][j]*B[i*COORD_DIM+1][j];
            a_dot_b += A[i*COORD_DIM+2][j]*B[i*COORD_DIM+2][j];
            AdotB[i][j] = a_dot_b;
          }
        }
        return AdotB;
      };
      auto compute_inner_prod = [&S, &area_elem](const Vector<ElemBasis>& A, const Vector<ElemBasis>& B) {
        const auto& quad_wts = ElemBasis::QuadWts();
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();
        const Long dof = B.Dim() / Nelem;
        Real sum = 0;
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            Real AdotB = 0;
            for (Long k = 0; k < dof; k++) {
              AdotB += A[i*dof+k][j] * B[i*dof+k][j];
            }
            sum += AdotB * area_elem[i][j] * quad_wts[j];
          }
        }
        return sum;
      };
      auto compute_norm_area_elem = [&S](Vector<ElemBasis>& normal, Vector<ElemBasis>& area_elem){ // Set normal, area_elem
        const Vector<ElemBasis> X = S.GetElemList().ElemVector();
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> dX;
        ElemBasis::Grad(dX, X);

        area_elem.ReInit(Nelem);
        normal.ReInit(Nelem * COORD_DIM);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            Tensor<Real,true,COORD_DIM> x, n;
            Tensor<Real,true,COORD_DIM,2> dx;
            x(0) = X[i*COORD_DIM+0][j];
            x(1) = X[i*COORD_DIM+1][j];
            x(2) = X[i*COORD_DIM+2][j];

            dx(0,0) = dX[i*COORD_DIM*2+0][j];
            dx(0,1) = dX[i*COORD_DIM*2+1][j];
            dx(1,0) = dX[i*COORD_DIM*2+2][j];
            dx(1,1) = dX[i*COORD_DIM*2+3][j];
            dx(2,0) = dX[i*COORD_DIM*2+4][j];
            dx(2,1) = dX[i*COORD_DIM*2+5][j];

            n(0) = dx(1,0) * dx(2,1) - dx(2,0) * dx(1,1);
            n(1) = dx(2,0) * dx(0,1) - dx(0,0) * dx(2,1);
            n(2) = dx(0,0) * dx(1,1) - dx(1,0) * dx(0,1);
            Real area_elem_ = sqrt<Real>(n(0)*n(0) + n(1)*n(1) + n(2)*n(2));
            Real ooae = 1 / area_elem_;
            n(0) *= ooae;
            n(1) *= ooae;
            n(2) *= ooae;

            normal[i*COORD_DIM+0][j] = n(0);
            normal[i*COORD_DIM+1][j] = n(1);
            normal[i*COORD_DIM+2][j] = n(2);
            area_elem[i][j] = area_elem_;
          }
        }
      };
      compute_norm_area_elem(normal, area_elem);

      S.quadrature_FxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxU , order_singular, order_direct, -1.0, comm);
      S.quadrature_DxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_DxU , order_singular, order_direct, -1.0, comm);
      S.quadrature_FxdU.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxdU, order_singular, order_direct, -1.0, comm);
      S.quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm);

      auto compute_poloidal_circulation = [&S] (const Vector<ElemBasis>& B) {
        const Vector<ElemBasis> X = S.GetElemList().ElemVector();
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();
        const auto& quad_wts = Basis<Real,1,ORDER>::QuadWts();

        Vector<ElemBasis> dX;
        ElemBasis::Grad(dX, X);
        const Long Nt = 40;
        const Long Np = 8;
        for (Long t = 0; t < Nt; t++) {
          for (Long j = 0; j < ORDER; j++) {
            Real sum = 0;
            for (Long p = 0; p < Np; p++) {
              for (Long i = 0; i < ORDER; i++) {
                Long elem_idx = t*Np+p;
                Long node_idx = i*ORDER+j;

                Tensor<Real,true,COORD_DIM,2> dx;
                dx(0,0) = dX[elem_idx*COORD_DIM*2+0][node_idx];
                dx(0,1) = dX[elem_idx*COORD_DIM*2+1][node_idx];
                dx(1,0) = dX[elem_idx*COORD_DIM*2+2][node_idx];
                dx(1,1) = dX[elem_idx*COORD_DIM*2+3][node_idx];
                dx(2,0) = dX[elem_idx*COORD_DIM*2+4][node_idx];
                dx(2,1) = dX[elem_idx*COORD_DIM*2+5][node_idx];

                Tensor<Real,true,COORD_DIM> b;
                b(0) = B[elem_idx*COORD_DIM+0][node_idx];
                b(1) = B[elem_idx*COORD_DIM+1][node_idx];
                b(2) = B[elem_idx*COORD_DIM+2][node_idx];

                sum += (b(0)*dx(0,1) + b(1)*dx(1,1) + b(2)*dx(2,1)) * quad_wts[i];
              }
            }
            std::cout<<sum<<' ';
          }
        }
        std::cout<<'\n';
      };
      auto compute_toroidal_circulation = [&S] (const Vector<ElemBasis>& B) {
        const Vector<ElemBasis> X = S.GetElemList().ElemVector();
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();
        const auto& quad_wts = Basis<Real,1,ORDER>::QuadWts();

        Vector<ElemBasis> dX;
        ElemBasis::Grad(dX, X);
        const Long Nt = 40;
        const Long Np = 8;
        for (Long p = 0; p < Np; p++) {
          for (Long i = 0; i < ORDER; i++) {
            Real sum = 0;
            for (Long t = 0; t < Nt; t++) {
              for (Long j = 0; j < ORDER; j++) {
                Long elem_idx = t*Np+p;
                Long node_idx = i*ORDER+j;

                Tensor<Real,true,COORD_DIM,2> dx;
                dx(0,0) = dX[elem_idx*COORD_DIM*2+0][node_idx];
                dx(0,1) = dX[elem_idx*COORD_DIM*2+1][node_idx];
                dx(1,0) = dX[elem_idx*COORD_DIM*2+2][node_idx];
                dx(1,1) = dX[elem_idx*COORD_DIM*2+3][node_idx];
                dx(2,0) = dX[elem_idx*COORD_DIM*2+4][node_idx];
                dx(2,1) = dX[elem_idx*COORD_DIM*2+5][node_idx];

                Tensor<Real,true,COORD_DIM> b;
                b(0) = B[elem_idx*COORD_DIM+0][node_idx];
                b(1) = B[elem_idx*COORD_DIM+1][node_idx];
                b(2) = B[elem_idx*COORD_DIM+2][node_idx];

                sum += (b(0)*dx(0,0) + b(1)*dx(1,0) + b(2)*dx(2,0)) * quad_wts[j];
              }
            }
            std::cout<<sum<<' ';
          }
        }
        std::cout<<'\n';
      };

      auto compute_poloidal_circulation_ = [&S,&area_elem] (const Vector<ElemBasis>& B) {
        const Vector<ElemBasis> X = S.GetElemList().ElemVector();
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();
        const auto& quad_wts = ElemBasis::QuadWts();

        Vector<ElemBasis> dX;
        ElemBasis::Grad(dX, X);
        Real sum = 0;
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            Tensor<Real,true,COORD_DIM,2> dx;
            dx(0,0) = dX[i*COORD_DIM*2+0][j];
            dx(0,1) = dX[i*COORD_DIM*2+1][j];
            dx(1,0) = dX[i*COORD_DIM*2+2][j];
            dx(1,1) = dX[i*COORD_DIM*2+3][j];
            dx(2,0) = dX[i*COORD_DIM*2+4][j];
            dx(2,1) = dX[i*COORD_DIM*2+5][j];

            Tensor<Real,true,COORD_DIM> b;
            b(0) = B[i*COORD_DIM+0][j];
            b(1) = B[i*COORD_DIM+1][j];
            b(2) = B[i*COORD_DIM+2][j];

            Real s = 1/area_elem[i][j];
            sum += (b(0)*dx(0,1) + b(1)*dx(1,1) + b(2)*dx(2,1)) * s * area_elem[i][j] * quad_wts[j];
          }
        }
        return sum;
      };
      auto compute_toroidal_circulation_ = [&S,&area_elem] (const Vector<ElemBasis>& B) {
        const Vector<ElemBasis> X = S.GetElemList().ElemVector();
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();
        const auto& quad_wts = ElemBasis::QuadWts();

        Vector<ElemBasis> dX;
        ElemBasis::Grad(dX, X);
        Real sum = 0;
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            Tensor<Real,true,COORD_DIM,2> dx;
            dx(0,0) = dX[i*COORD_DIM*2+0][j];
            dx(0,1) = dX[i*COORD_DIM*2+1][j];
            dx(1,0) = dX[i*COORD_DIM*2+2][j];
            dx(1,1) = dX[i*COORD_DIM*2+3][j];
            dx(2,0) = dX[i*COORD_DIM*2+4][j];
            dx(2,1) = dX[i*COORD_DIM*2+5][j];

            Tensor<Real,true,COORD_DIM> b;
            b(0) = B[i*COORD_DIM+0][j];
            b(1) = B[i*COORD_DIM+1][j];
            b(2) = B[i*COORD_DIM+2][j];

            Real s = 1/area_elem[i][j];
            sum += (b(0)*dx(0,0) + b(1)*dx(1,0) + b(2)*dx(2,0)) * s * area_elem[i][j] * quad_wts[j];
          }
        }
        return sum;
      };

      auto compute_grad_adj = [&S,&area_elem] (const Vector<ElemBasis>& V) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> du_dX(Nelem*COORD_DIM*2);
        { // Set du_dX
          Vector<ElemBasis> dX;
          ElemBasis::Grad(dX, S.GetElemList().ElemVector());

          auto inv2x2 = [](Tensor<Real, true, 2, 2> M) {
            Tensor<Real, true, 2, 2> Mout;
            Real oodet = 1 / (M(0,0) * M(1,1) - M(0,1) * M(1,0));
            Mout(0,0) =  M(1,1) * oodet;
            Mout(0,1) = -M(0,1) * oodet;
            Mout(1,0) = -M(1,0) * oodet;
            Mout(1,1) =  M(0,0) * oodet;
            return Mout;
          };
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Tensor<Real, true, 3, 2> dX_du;
              dX_du(0,0) = dX[(i*COORD_DIM+0)*2+0][j];
              dX_du(1,0) = dX[(i*COORD_DIM+1)*2+0][j];
              dX_du(2,0) = dX[(i*COORD_DIM+2)*2+0][j];
              dX_du(0,1) = dX[(i*COORD_DIM+0)*2+1][j];
              dX_du(1,1) = dX[(i*COORD_DIM+1)*2+1][j];
              dX_du(2,1) = dX[(i*COORD_DIM+2)*2+1][j];

              Tensor<Real, true, 2, 2> G; // = dX_du.Transpose() * dX_du;
              G(0,0) = dX_du(0,0) * dX_du(0,0) + dX_du(1,0) * dX_du(1,0) + dX_du(2,0) * dX_du(2,0);
              G(0,1) = dX_du(0,0) * dX_du(0,1) + dX_du(1,0) * dX_du(1,1) + dX_du(2,0) * dX_du(2,1);
              G(1,0) = dX_du(0,1) * dX_du(0,0) + dX_du(1,1) * dX_du(1,0) + dX_du(2,1) * dX_du(2,0);
              G(1,1) = dX_du(0,1) * dX_du(0,1) + dX_du(1,1) * dX_du(1,1) + dX_du(2,1) * dX_du(2,1);

              Tensor<Real, true, 2, 2> Ginv = inv2x2(G);
              du_dX[(i*COORD_DIM+0)*2+0][j] = Ginv(0,0) * dX_du(0,0) + Ginv(0,1) * dX_du(0,1);
              du_dX[(i*COORD_DIM+1)*2+0][j] = Ginv(0,0) * dX_du(1,0) + Ginv(0,1) * dX_du(1,1);
              du_dX[(i*COORD_DIM+2)*2+0][j] = Ginv(0,0) * dX_du(2,0) + Ginv(0,1) * dX_du(2,1);
              du_dX[(i*COORD_DIM+0)*2+1][j] = Ginv(1,0) * dX_du(0,0) + Ginv(1,1) * dX_du(0,1);
              du_dX[(i*COORD_DIM+1)*2+1][j] = Ginv(1,0) * dX_du(1,0) + Ginv(1,1) * dX_du(1,1);
              du_dX[(i*COORD_DIM+2)*2+1][j] = Ginv(1,0) * dX_du(2,0) + Ginv(1,1) * dX_du(2,1);
            }
          }
        }

        Vector<ElemBasis> dudX_V(Nelem*2);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            dudX_V[i*2+0][j] = 0;
            dudX_V[i*2+1][j] = 0;

            dudX_V[i*2+0][j] += du_dX[(i*COORD_DIM+0)*2+0][j] * V[i*COORD_DIM+0][j];
            dudX_V[i*2+0][j] += du_dX[(i*COORD_DIM+1)*2+0][j] * V[i*COORD_DIM+1][j];
            dudX_V[i*2+0][j] += du_dX[(i*COORD_DIM+2)*2+0][j] * V[i*COORD_DIM+2][j];

            dudX_V[i*2+1][j] += du_dX[(i*COORD_DIM+0)*2+1][j] * V[i*COORD_DIM+0][j];
            dudX_V[i*2+1][j] += du_dX[(i*COORD_DIM+1)*2+1][j] * V[i*COORD_DIM+1][j];
            dudX_V[i*2+1][j] += du_dX[(i*COORD_DIM+2)*2+1][j] * V[i*COORD_DIM+2][j];
          }
        }

        Vector<ElemBasis> eye(Nnodes), Mgrad;
        eye = 0;
        for (Long i = 0; i < Nnodes; i++) eye[i][i] = 1;
        ElemBasis::Grad(Mgrad, eye);

        Vector<ElemBasis> grad_adj_V(Nelem);
        const auto& quad_wts = ElemBasis::QuadWts();
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            Real sum = 0;
            for (Long k = 0; k < Nnodes; k++) {
              sum += Mgrad[j*2+0][k] * dudX_V[i*2+0][k] * (area_elem[i][k] * quad_wts[k]) / (quad_wts[j] * area_elem[i][j]);
              sum += Mgrad[j*2+1][k] * dudX_V[i*2+1][k] * (area_elem[i][k] * quad_wts[k]) / (quad_wts[j] * area_elem[i][j]);
            }
            grad_adj_V[i][j] = -sum;
          }
        }
        return grad_adj_V;
      };

      auto compute_B0 = [&S](const Real alpha) { // alpha/|r| \hat{\theta}
        const Vector<ElemBasis> X = S.GetElemList().ElemVector();
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> B0(Nelem * COORD_DIM);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            Tensor<Real,true,COORD_DIM> x, b0, axis;
            x(0) = X[i*COORD_DIM+0][j];
            x(1) = X[i*COORD_DIM+1][j];
            x(2) = X[i*COORD_DIM+2][j];

            axis(0) = 0;
            axis(1) = 0;
            axis(2) = 1;
            b0(0) = axis(1) * x(2) - axis(2) * x(1);
            b0(1) = axis(2) * x(0) - axis(0) * x(2);
            b0(2) = axis(0) * x(1) - axis(1) * x(0);
            Real scale = 1 / (b0(0)*b0(0) + b0(1)*b0(1) + b0(2)*b0(2));
            b0(0) *= scale;
            b0(1) *= scale;
            b0(2) *= scale;

            B0[i*COORD_DIM+0][j] = alpha * b0(0);
            B0[i*COORD_DIM+1][j] = alpha * b0(1);
            B0[i*COORD_DIM+2][j] = alpha * b0(2);
          }
        }
        return B0;
      };
      auto compute_dB0 = [&S](const Real alpha) { // alpha/|r| \hat{\theta}
        const Vector<ElemBasis> X = S.GetElemList().ElemVector();
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> dB0(Nelem * COORD_DIM * COORD_DIM);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            Tensor<Real,true,COORD_DIM> x;
            x(0) = X[i*COORD_DIM+0][j];
            x(1) = X[i*COORD_DIM+1][j];
            x(2) = X[i*COORD_DIM+2][j];
            Real R2inv = 1 / (x(0)*x(0) + x(1)*x(1));

            dB0[(i*COORD_DIM+0)*COORD_DIM+0][j] = alpha * (2*x(0)*x(1) * R2inv*R2inv);
            dB0[(i*COORD_DIM+0)*COORD_DIM+1][j] = alpha * (-R2inv + 2*x(1)*x(1) * R2inv*R2inv);
            dB0[(i*COORD_DIM+0)*COORD_DIM+2][j] = 0;

            dB0[(i*COORD_DIM+1)*COORD_DIM+0][j] = alpha * (R2inv - 2*x(0)*x(0) * R2inv*R2inv);
            dB0[(i*COORD_DIM+1)*COORD_DIM+1][j] = alpha * (-2*x(0)*x(1) * R2inv*R2inv);
            dB0[(i*COORD_DIM+1)*COORD_DIM+2][j] = 0;

            dB0[(i*COORD_DIM+2)*COORD_DIM+0][j] = 0;
            dB0[(i*COORD_DIM+2)*COORD_DIM+1][j] = 0;
            dB0[(i*COORD_DIM+2)*COORD_DIM+2][j] = 0;
          }
        }
        return dB0;
      };
      auto compute_half_n_plus_dG = [&S, &normal](const Vector<ElemBasis>& sigma) { // B = n sigma/2 + dG[sigma]
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> B;
        S.quadrature_FxdU.Eval(B, S.GetElemList(), sigma, S.Laplace_FxdU);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            for (Long k = 0; k < COORD_DIM; k++) {
              B[i*COORD_DIM+k][j] -= 0.5*sigma[i][j]*normal[i*COORD_DIM+k][j];
            }
          }
        }
        return B;
      };

      auto compute_A11 = [&S,&normal,&compute_half_n_plus_dG,&compute_dot_prod](Vector<Real>& B_dot_n, const Vector<Real>& sigma) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();
        B_dot_n.ReInit(Nelem * Nnodes);
        Vector<ElemBasis> sigma_(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            sigma_[i][j] = sigma[i*Nnodes+j];
          }
        }
        Vector<ElemBasis> B_dot_n_ = compute_dot_prod(normal, compute_half_n_plus_dG(sigma_));
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            B_dot_n[i*Nnodes+j] = B_dot_n_[i][j];
          }
        }
      };
      auto compute_A12 = [&S,&normal,&compute_dot_prod,&compute_B0](Vector<Real>& B_dot_n, const Real alpha) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();
        B_dot_n.ReInit(Nelem * Nnodes);
        Vector<ElemBasis> B_dot_n_ = compute_dot_prod(normal, compute_B0(alpha));
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            B_dot_n[i*Nnodes+j] = B_dot_n_[i][j];
          }
        }
      };
      auto compute_A21 = [&S,&normal,&compute_half_n_plus_dG,&compute_poloidal_circulation_](const Vector<Real>& sigma) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> sigma_(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            sigma_[i][j] = sigma[i*Nnodes+j];
          }
        }

        if (0) { // alternate implementation
          //Vector<ElemBasis> A21_(Nelem);
          //Vector<Real> A21(Nelem*Nnodes);
          //compute_A21adj(A21, 1);
          //for (Long i = 0; i < Nelem; i++) {
          //  for (Long j = 0; j < Nnodes; j++) {
          //    A21_[i][j] = A21[i*Nnodes+j];
          //  }
          //}
          //return compute_inner_prod(A21_, sigma_);
        }
        Vector<ElemBasis> B = compute_half_n_plus_dG(sigma_);

        Vector<ElemBasis> J(Nelem * COORD_DIM);
        for (Long i = 0; i < Nelem; i++) { // Set J
          for (Long j = 0; j < Nnodes; j++) {
            Tensor<Real,true,COORD_DIM> b, n;
            b(0) = B[i*COORD_DIM+0][j];
            b(1) = B[i*COORD_DIM+1][j];
            b(2) = B[i*COORD_DIM+2][j];

            n(0) = normal[i*COORD_DIM+0][j];
            n(1) = normal[i*COORD_DIM+1][j];
            n(2) = normal[i*COORD_DIM+2][j];

            J[i*COORD_DIM+0][j] = n(1) * b(2) - n(2) * b(1);
            J[i*COORD_DIM+1][j] = n(2) * b(0) - n(0) * b(2);
            J[i*COORD_DIM+2][j] = n(0) * b(1) - n(1) * b(0);
          }
        }

        Vector<ElemBasis> A;
        S.quadrature_FxU.Eval(A, S.GetElemList(), J, S.Laplace_FxU);
        return compute_poloidal_circulation_(A)/S.NtNp_[0];
      };
      auto compute_A22 = [&S,&compute_B0,&normal,&compute_poloidal_circulation_](const Real alpha) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> B = compute_B0(alpha);

        Vector<ElemBasis> J(Nelem * COORD_DIM);
        for (Long i = 0; i < Nelem; i++) { // Set J
          for (Long j = 0; j < Nnodes; j++) {
            Tensor<Real,true,COORD_DIM> b, n;
            b(0) = B[i*COORD_DIM+0][j];
            b(1) = B[i*COORD_DIM+1][j];
            b(2) = B[i*COORD_DIM+2][j];

            n(0) = normal[i*COORD_DIM+0][j];
            n(1) = normal[i*COORD_DIM+1][j];
            n(2) = normal[i*COORD_DIM+2][j];

            J[i*COORD_DIM+0][j] = n(1) * b(2) - n(2) * b(1);
            J[i*COORD_DIM+1][j] = n(2) * b(0) - n(0) * b(2);
            J[i*COORD_DIM+2][j] = n(0) * b(1) - n(1) * b(0);
          }
        }

        Vector<ElemBasis> A;
        S.quadrature_FxU.Eval(A, S.GetElemList(), J, S.Laplace_FxU);
        return compute_poloidal_circulation_(A)/S.NtNp_[0];
      };
      auto compute_A = [&compute_A11,&compute_A12,&compute_A21,&compute_A22] (const Vector<Real>& x) {
        const Vector<Real> sigma(x.Dim()-1,(Iterator<Real>)x.begin(),false);
        const Real& alpha = x[x.Dim()-1];

        Vector<Real> Ax;
        Ax.ReInit(x.Dim());
        Vector<Real> Bdotn(x.Dim()-1,Ax.begin(),false);
        Real& flux = Ax[x.Dim()-1];

        Vector<Real> Adotn_0, Adotn_1;
        compute_A11(Adotn_0, sigma);
        compute_A12(Adotn_1, alpha);
        Bdotn = Adotn_0 + Adotn_1;

        flux = compute_A21(sigma) + compute_A22(alpha);
        return Ax;
      };
      auto compute_invA = [&S,&comm,&compute_A] (Vector<ElemBasis>& sigma, Real& alpha, Real flux) {
        typename sctl::ParallelSolver<Real>::ParallelOp BIOp = [&compute_A](sctl::Vector<Real>* Ax, const sctl::Vector<Real>& x) {
          (*Ax) = compute_A(x);
        };
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<Real> rhs_(Nelem * Nnodes + 1);
        rhs_ = 0;
        rhs_[Nelem * Nnodes] = flux;

        Vector<Real> x_(Nelem * Nnodes + 1);
        x_ = 0;
        ParallelSolver<Real> linear_solver(comm, true);
        linear_solver(&x_, BIOp, rhs_, 1e-8, 50);

        sigma.ReInit(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            sigma[i][j] = x_[i*Nnodes+j];
          }
        }
        alpha = x_[Nelem * Nnodes];
      };
      auto compute_invA_ = [&S,&comm,&compute_A] (Vector<Real>& b) {
        typename sctl::ParallelSolver<Real>::ParallelOp BIOp = [&compute_A](sctl::Vector<Real>* Ax, const sctl::Vector<Real>& x) {
          (*Ax) = compute_A(x);
        };
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<Real> x(b.Dim());
        x = 0;
        ParallelSolver<Real> linear_solver(comm, true);
        linear_solver(&x, BIOp, b, 1e-8, 50);
        return x;
      };

      auto compute_A11adj = [&S](Vector<Real>& U, const Vector<Real>& sigma) { // A11adj  = I/2 + D
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> sigma_(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            sigma_[i][j] = sigma[i*Nnodes+j];
          }
        }
        S.quadrature_DxU.Eval(U, S.GetElemList(), sigma_, S.Laplace_DxU);
        U = sigma*(-0.5) + U;
      };
      auto compute_A12adj = [&S,&compute_A12,&compute_inner_prod](const Vector<Real>& sigma_) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<Real> A12_sigma_;
        compute_A12(A12_sigma_, 1);

        Vector<ElemBasis> A12_sigma(Nelem), sigma(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            sigma[i][j] = sigma_[i*Nnodes+j];
            A12_sigma[i][j] = A12_sigma_[i*Nnodes+j];
          }
        }
        return compute_inner_prod(A12_sigma, sigma);
      };
      auto compute_A21adj = [&S,&area_elem,&normal](Vector<Real>& A21adj_flux, Real flux) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> density(Nelem * COORD_DIM);
        { // Set density
          Vector<ElemBasis> dX;
          ElemBasis::Grad(dX, S.GetElemList().ElemVector());
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Tensor<Real,true,COORD_DIM,2> dx;
              dx(0,0) = dX[i*COORD_DIM*2+0][j];
              dx(0,1) = dX[i*COORD_DIM*2+1][j];
              dx(1,0) = dX[i*COORD_DIM*2+2][j];
              dx(1,1) = dX[i*COORD_DIM*2+3][j];
              dx(2,0) = dX[i*COORD_DIM*2+4][j];
              dx(2,1) = dX[i*COORD_DIM*2+5][j];

              Real s = 1 / (area_elem[i][j] * S.NtNp_[0]);
              for (Long k = 0; k < COORD_DIM; k++) {
                density[i*COORD_DIM+k][j] = dx(k,1) * s;
              }
            }
          }
        }

        Vector<ElemBasis> Gdensity;
        S.quadrature_FxU.Eval(Gdensity, S.GetElemList(), density, S.Laplace_FxU);

        Vector<ElemBasis> nxGdensity(Nelem * COORD_DIM);
        for (Long i = 0; i < Nelem; i++) { // Set nxGdensity
          for (Long j = 0; j < Nnodes; j++) {
            Tensor<Real,true,COORD_DIM> Gdensity_, n;
            Gdensity_(0) = Gdensity[i*COORD_DIM+0][j];
            Gdensity_(1) = Gdensity[i*COORD_DIM+1][j];
            Gdensity_(2) = Gdensity[i*COORD_DIM+2][j];

            n(0) = normal[i*COORD_DIM+0][j];
            n(1) = normal[i*COORD_DIM+1][j];
            n(2) = normal[i*COORD_DIM+2][j];

            nxGdensity[i*COORD_DIM+0][j] = n(1) * Gdensity_(2) - n(2) * Gdensity_(1);
            nxGdensity[i*COORD_DIM+1][j] = n(2) * Gdensity_(0) - n(0) * Gdensity_(2);
            nxGdensity[i*COORD_DIM+2][j] = n(0) * Gdensity_(1) - n(1) * Gdensity_(0);
          }
        }
        S.quadrature_dUxF.Eval(A21adj_flux, S.GetElemList(), nxGdensity, S.Laplace_dUxF);
        A21adj_flux *= flux;
      };
      auto compute_A22adj = [&compute_A22] (const Real alpha) {
        return compute_A22(alpha);
      };
      auto compute_Aadj = [&compute_A11adj,&compute_A12adj,&compute_A21adj,&compute_A22adj] (const Vector<Real>& x) {
        const Vector<Real> sigma(x.Dim()-1,(Iterator<Real>)x.begin(),false);
        const Real& alpha = x[x.Dim()-1];

        Vector<Real> Ax;
        Ax.ReInit(x.Dim());
        Vector<Real> Bdotn(x.Dim()-1,Ax.begin(),false);
        Real& flux = Ax[x.Dim()-1];

        Vector<Real> Adotn_0, Adotn_1;
        compute_A11adj(Adotn_0, sigma);
        compute_A21adj(Adotn_1, alpha);
        Bdotn = Adotn_0 + Adotn_1;

        flux = compute_A12adj(sigma) + compute_A22adj(alpha);
        return Ax;
      };
      auto compute_invAadj = [&S,&comm,&compute_Aadj] (Vector<Real>& b) {
        typename sctl::ParallelSolver<Real>::ParallelOp BIOp = [&compute_Aadj](sctl::Vector<Real>* Ax, const sctl::Vector<Real>& x) {
          (*Ax) = compute_Aadj(x);
        };
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<Real> x(b.Dim());
        x = 0;
        ParallelSolver<Real> linear_solver(comm, true);
        linear_solver(&x, BIOp, b, 1e-8, 50);
        return x;
      };

      auto compute_dg_dsigma = [&S, &normal, &compute_dot_prod](const Vector<ElemBasis>& B) { // dg_dsigma = \int 2 B \cdot (\nabla G + n/2)
        Vector<ElemBasis> B_dot_gradG;
        S.quadrature_dUxF.Eval(B_dot_gradG, S.GetElemList(), B, S.Laplace_dUxF);
        return B_dot_gradG * (-2.0) + compute_dot_prod(B,normal);
      };
      auto compute_dg_dalpha = [&S,&compute_B0,&compute_inner_prod] (const Vector<ElemBasis>& B) {
        auto dB_dalpha = compute_B0(1);
        return 2*compute_inner_prod(B,dB_dalpha);
      };

      auto compute_dg_dnu = [&S,&comm,&normal,&compute_inner_prod,&area_elem,&compute_dB0](const Vector<ElemBasis>& sigma, Real alpha, const Vector<ElemBasis>& B) { // dg_dnu = (B*B) 2H - (2 B) \cdot (n \cdnot nabla) \nabla G[sigma] + (2 B) \alpha dB0_dnu \hat{\theta} + sigma (\nabla D)^T [2 B] + (2H) sigma (\nabla G)^T [2 B]
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();
        Vector<ElemBasis> v = B * 2.0;

        Vector<ElemBasis> dg_dnu0(Nelem), dg_dnu1(Nelem), dg_dnu2(Nelem), dg_dnu3(Nelem), dg_dnu4(Nelem);
        dg_dnu0 = 0;
        dg_dnu1 = 0;
        dg_dnu2 = 0;
        dg_dnu3 = 0;
        dg_dnu4 = 0;

        Vector<ElemBasis> H(Nelem);
        { // Set mean curvature H
          const Vector<ElemBasis> X = S.GetElemList().ElemVector();
          Vector<ElemBasis> dX, d2X;
          ElemBasis::Grad(dX, X);
          ElemBasis::Grad(d2X, dX);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Tensor<Real,true,2,2> I, invI, II;
              for (Long k0 = 0; k0 < 2; k0++) {
                for (Long k1 = 0; k1 < 2; k1++) {
                  I(k0,k1) = 0;
                  I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                  I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                  I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                  II(k0,k1) = 0;
                  II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                  II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                  II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                }
              }
              { // Set invI
                Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                invI(0,0) = I(1,1) / detI;
                invI(0,1) = -I(0,1) / detI;
                invI(1,0) = -I(1,0) / detI;
                invI(1,1) = I(0,0) / detI;
              }
              { // Set H
                H[i][j] = 0;
                H[i][j] += -0.5 * II(0,0)*invI(0,0);
                H[i][j] += -0.5 * II(0,1)*invI(0,1);
                H[i][j] += -0.5 * II(1,0)*invI(1,0);
                H[i][j] += -0.5 * II(1,1)*invI(1,1);
              }
            }
          }
        }

        // dg_dnu = (B*B) 2H
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            dg_dnu0[i][j] = 0;
            dg_dnu0[i][j] += B[i*COORD_DIM+0][j] * B[i*COORD_DIM+0][j] * (2.0*H[i][j]);
            dg_dnu0[i][j] += B[i*COORD_DIM+1][j] * B[i*COORD_DIM+1][j] * (2.0*H[i][j]);
            dg_dnu0[i][j] += B[i*COORD_DIM+2][j] * B[i*COORD_DIM+2][j] * (2.0*H[i][j]);
          }
        }

        // dg_dnu1 = (2 B) \cdot (n \cdnot nabla) \nabla G[sigma]
        Vector<ElemBasis> d2Gsigma;
        Quadrature<Real> quadrature_Fxd2U;
        quadrature_Fxd2U.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_Fxd2U, order_singular, order_direct, -1.0, comm, -0.01 * pow<-2,Real>(ORDER));
        quadrature_Fxd2U.Eval(d2Gsigma, S.GetElemList(), sigma, S.Laplace_Fxd2U);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            dg_dnu1[i][j] = 0;
            dg_dnu1[i][j] -= d2Gsigma[i*9+0][j] * normal[i*COORD_DIM+0][j] * v[i*COORD_DIM+0][j];
            dg_dnu1[i][j] -= d2Gsigma[i*9+1][j] * normal[i*COORD_DIM+0][j] * v[i*COORD_DIM+1][j];
            dg_dnu1[i][j] -= d2Gsigma[i*9+2][j] * normal[i*COORD_DIM+0][j] * v[i*COORD_DIM+2][j];

            dg_dnu1[i][j] -= d2Gsigma[i*9+3][j] * normal[i*COORD_DIM+1][j] * v[i*COORD_DIM+0][j];
            dg_dnu1[i][j] -= d2Gsigma[i*9+4][j] * normal[i*COORD_DIM+1][j] * v[i*COORD_DIM+1][j];
            dg_dnu1[i][j] -= d2Gsigma[i*9+5][j] * normal[i*COORD_DIM+1][j] * v[i*COORD_DIM+2][j];

            dg_dnu1[i][j] -= d2Gsigma[i*9+6][j] * normal[i*COORD_DIM+2][j] * v[i*COORD_DIM+0][j];
            dg_dnu1[i][j] -= d2Gsigma[i*9+7][j] * normal[i*COORD_DIM+2][j] * v[i*COORD_DIM+1][j];
            dg_dnu1[i][j] -= d2Gsigma[i*9+8][j] * normal[i*COORD_DIM+2][j] * v[i*COORD_DIM+2][j];
          }
        }

        // dg_dnu2 = (2 B) \alpha dB0_dnu \hat{\theta}
        Vector<ElemBasis> dB0 = compute_dB0(alpha);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            dg_dnu2[i][j] = 0;
            dg_dnu2[i][j] += dB0[i*9+0][j] * normal[i*COORD_DIM+0][j] * v[i*COORD_DIM+0][j];
            dg_dnu2[i][j] += dB0[i*9+1][j] * normal[i*COORD_DIM+1][j] * v[i*COORD_DIM+0][j];
            dg_dnu2[i][j] += dB0[i*9+2][j] * normal[i*COORD_DIM+2][j] * v[i*COORD_DIM+0][j];

            dg_dnu2[i][j] += dB0[i*9+3][j] * normal[i*COORD_DIM+0][j] * v[i*COORD_DIM+1][j];
            dg_dnu2[i][j] += dB0[i*9+4][j] * normal[i*COORD_DIM+1][j] * v[i*COORD_DIM+1][j];
            dg_dnu2[i][j] += dB0[i*9+5][j] * normal[i*COORD_DIM+2][j] * v[i*COORD_DIM+1][j];

            dg_dnu2[i][j] += dB0[i*9+6][j] * normal[i*COORD_DIM+0][j] * v[i*COORD_DIM+2][j];
            dg_dnu2[i][j] += dB0[i*9+7][j] * normal[i*COORD_DIM+1][j] * v[i*COORD_DIM+2][j];
            dg_dnu2[i][j] += dB0[i*9+8][j] * normal[i*COORD_DIM+2][j] * v[i*COORD_DIM+2][j];
          }
        }

        // dg_dnu3 = (sigma (\nabla D)^T [2 B]
        Vector<ElemBasis> nablaDtv;
        Quadrature<Real> quadrature_dUxD;
        quadrature_dUxD.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxD, order_singular, order_direct, -1.0, comm, 0.01 * pow<-2,Real>(ORDER));
        quadrature_dUxD.Eval(nablaDtv, S.GetElemList(), v, S.Laplace_dUxD);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            dg_dnu3[i][j] = 0;
            dg_dnu3[i][j] += sigma[i][j] * nablaDtv[i*COORD_DIM+0][j]*normal[i*COORD_DIM+0][j];
            dg_dnu3[i][j] += sigma[i][j] * nablaDtv[i*COORD_DIM+1][j]*normal[i*COORD_DIM+1][j];
            dg_dnu3[i][j] += sigma[i][j] * nablaDtv[i*COORD_DIM+2][j]*normal[i*COORD_DIM+2][j];
          }
        }

        // dg_dnu4 = (2H) sigma (\nabla G)^T [2 B]
        Quadrature<Real> quadrature_dUxF;
        quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm, 0.01 * pow<-2,Real>(ORDER));
        quadrature_dUxF.Eval(dg_dnu4, S.GetElemList(), v, S.Laplace_dUxF);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            dg_dnu4[i][j] *= 2*H[i][j] * sigma[i][j];
          }
        }

        return dg_dnu0 + dg_dnu1 + dg_dnu2 + dg_dnu3 - dg_dnu4;
      };

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      Real flux = 1.0, alpha;
      Vector<ElemBasis> sigma(S.GetElemList().NElem());
      compute_invA(sigma, alpha, flux);
      Vector<ElemBasis> B = compute_half_n_plus_dG(sigma) + compute_B0(alpha);
      Real g = compute_inner_prod(B, B);
      std::cout<<"g = "<<g<<'\n';

      { // Write VTU
        VTUData vtu;
        vtu.AddElems(S.GetElemList(), sigma, ORDER);
        vtu.WriteVTK("sigma", comm);
      }
      { // Write VTU
        VTUData vtu;
        vtu.AddElems(S.GetElemList(), B, ORDER);
        vtu.WriteVTK("B", comm);
      }

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      if (0) { // test dg_dnu
        auto compute_g = [&S,&comm,&normal,&area_elem,&sigma,&alpha,&compute_norm_area_elem,&compute_B0,&compute_inner_prod](const Vector<ElemBasis>& nu, Real eps) {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> X_orig(Nelem*COORD_DIM);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              X_orig[i*COORD_DIM+0][j] = S.Elem(i,0)[j];
              X_orig[i*COORD_DIM+1][j] = S.Elem(i,1)[j];
              X_orig[i*COORD_DIM+2][j] = S.Elem(i,2)[j];
              S.Elem(i,0)[j] += eps*nu[i][j] * normal[i*COORD_DIM+0][j];
              S.Elem(i,1)[j] += eps*nu[i][j] * normal[i*COORD_DIM+1][j];
              S.Elem(i,2)[j] += eps*nu[i][j] * normal[i*COORD_DIM+2][j];
            }
          }
          compute_norm_area_elem(normal, area_elem);

          Vector<Real> Xt(Nelem*Nnodes*COORD_DIM);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              for (Long k = 0; k < COORD_DIM; k++) {
                Xt[(i*Nnodes+j)*COORD_DIM+k] = S.Elem(i,k)[j] - 1e-4*normal[i*COORD_DIM+k][j];// + eps*nu[i][j] * normal[i*COORD_DIM+k][j];
              }
            }
          }
          Vector<ElemBasis> B0 = compute_B0(alpha);

          Vector<ElemBasis> B1;
          Quadrature<Real> quadrature_FxdU;
          quadrature_FxdU.template Setup<ElemBasis>(S.GetElemList(), Xt, S.Laplace_FxdU, order_singular, order_direct, -1, comm);
          quadrature_FxdU.Eval(B1, S.GetElemList(), sigma, S.Laplace_FxdU);

          Real g = compute_inner_prod(B0+B1, B0+B1);

          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              S.Elem(i,0)[j] = X_orig[i*COORD_DIM+0][j];
              S.Elem(i,1)[j] = X_orig[i*COORD_DIM+1][j];
              S.Elem(i,2)[j] = X_orig[i*COORD_DIM+2][j];
            }
          }
          compute_norm_area_elem(normal, area_elem);

          return g;
        };

        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> nu(Nelem);
        nu = 1; //area_elem;

        Vector<ElemBasis> dg_dnu = compute_dg_dnu(sigma, alpha, B);
        std::cout<<compute_inner_prod(dg_dnu, nu)<<'\n';
        { // Write VTU
          VTUData vtu;
          vtu.AddElems(S.GetElemList(), dg_dnu, ORDER);
          vtu.WriteVTK("dg_dnu", comm);
        }

        Real eps = 1e-5;
        Real g0 = compute_g(nu,-eps);
        Real g1 = compute_g(nu,eps);
        std::cout<<"g = "<<g0<<"  g = "<<g1<<"  dg_dnu = "<<(g1-g0)/(2*eps)<<'\n';
      }

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      if (0) { // test dg_dsigma
        Vector<ElemBasis> dg_dsigma = compute_dg_dsigma(B);
        { // Write VTU
          VTUData vtu;
          vtu.AddElems(S.GetElemList(), dg_dsigma, ORDER);
          vtu.WriteVTK("dg_dsigma", comm);
        }

        Real dt = 1e-1;
        const Long Nelem = S.GetElemList().NElem();
        const auto& quad_wts = ElemBasis::QuadWts();

        Vector<ElemBasis> dg_dsigma_(Nelem);
        dg_dsigma_ = 0;
        for (Long i = 0; i < Nelem; i++) { // Set dg_dsigma_
          for (Long j = 0; j < ElemBasis::Size(); j++) {
            auto sigma_0 = sigma;
            auto sigma_1 = sigma;
            sigma_0[i][j] -= 0.5*dt;
            sigma_1[i][j] += 0.5*dt;
            auto B_0 = compute_half_n_plus_dG(sigma_0) + compute_B0(alpha);
            auto B_1 = compute_half_n_plus_dG(sigma_1) + compute_B0(alpha);
            auto g_0 = compute_inner_prod(B_0, B_0);
            auto g_1 = compute_inner_prod(B_1, B_1);
            dg_dsigma_[i][j] = (g_1 - g_0) / dt;
            dg_dsigma_[i][j] /= quad_wts[j] * area_elem[i][j];
            std::cout<<dg_dsigma_[i][j]<<' '<<j<<' '<<ElemBasis::Size()<<'\n'; ////////////////
          }
          { // Write VTU
            VTUData vtu;
            vtu.AddElems(S.GetElemList(), dg_dsigma_, ORDER);
            vtu.WriteVTK("dg_dsigma_", comm);
          }
        }
      }
      if (0) { // test dg_dalpha
        Real dg_dalpha = compute_dg_dalpha(B);

        Real dt = 1e-1;
        auto B_0 = compute_half_n_plus_dG(sigma) + compute_B0(alpha - 0.5*dt);
        auto B_1 = compute_half_n_plus_dG(sigma) + compute_B0(alpha + 0.5*dt);
        auto g_0 = compute_inner_prod(B_0, B_0);
        auto g_1 = compute_inner_prod(B_1, B_1);
        Real dg_dalpha_ = (g_1 - g_0) / dt;
        std::cout<<dg_dalpha<<' '<<dg_dalpha_<<'\n';
      }

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      if (0) { // test compute_A21adj
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<Real> A21adj_;
        compute_A21adj(A21adj_, flux);
        Vector<ElemBasis> A21adj(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            A21adj[i][j] = A21adj_[i*Nnodes+j];
          }
        }
        { // Write VTU
          VTUData vtu;
          vtu.AddElems(S.GetElemList(), A21adj, ORDER);
          vtu.WriteVTK("A21adj", comm);
        }

        { // verify
          Vector<Real> sigma_(Nelem*Nnodes);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              sigma_[i*Nnodes+j] = sigma[i][j];
            }
          }
          Real flux = compute_inner_prod(A21adj, sigma);
          std::cout<<"Error: "<<compute_A21(sigma_)-flux<<'\n';
        }
        { // compute finite-difference matrix
          Real dt = 1e+1;
          const Long Nelem = S.GetElemList().NElem();
          const auto& quad_wts = ElemBasis::QuadWts();

          Vector<ElemBasis> A21(Nelem);
          A21 = 0;
          for (Long i = 0; i < Nelem; i++) { // Set A21
            for (Long j = 0; j < ElemBasis::Size(); j++) {
              Vector<Real> sigma_0(Nelem*ElemBasis::Size());
              Vector<Real> sigma_1(Nelem*ElemBasis::Size());
              sigma_0 = 0;
              sigma_1 = 0;
              sigma_0[i*ElemBasis::Size()+j] -= 0.5*dt;
              sigma_1[i*ElemBasis::Size()+j] += 0.5*dt;
              auto flux_0 = compute_A21(sigma_0);
              auto flux_1 = compute_A21(sigma_1);
              A21[i][j] = (flux_1 - flux_0) / dt;
              A21[i][j] /= quad_wts[j] * area_elem[i][j];
              std::cout<<A21[i][j]<<' '<<j<<' '<<ElemBasis::Size()<<'\n'; ////////////////
            }
            { // Write VTU
              VTUData vtu;
              vtu.AddElems(S.GetElemList(), A21, ORDER);
              vtu.WriteVTK("A21", comm);
            }
          }
        }
      }

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      auto compute_invA11 = [&S,&normal,&comm,&compute_A11](const Vector<ElemBasis>& rhs) { // Solver for sigma: sigma/2 + n.dG[sigma] = rhs
        typename sctl::ParallelSolver<Real>::ParallelOp BIOp = [&S,&normal,&compute_A11](sctl::Vector<Real>* A11_sigma, const sctl::Vector<Real>& sigma) {
          compute_A11(*A11_sigma, sigma);
        };
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> sigma(Nelem);
        Vector<Real> rhs_(Nelem * Nnodes), sigma_(Nelem * Nnodes);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            rhs_[i*Nnodes+j] = rhs[i][j];
            sigma_[i*Nnodes+j] = 0;
          }
        }
        ParallelSolver<Real> linear_solver(comm, true);
        linear_solver(&sigma_, BIOp, rhs_, 1e-8, 50);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            sigma[i][j] = sigma_[i*Nnodes+j];
          }
        }
        return sigma;
      };
      auto compute_invA11adj = [&S,&normal,&comm,&compute_A11adj](const Vector<ElemBasis>& rhs) { // Solver for sigma: A11adj sigma = rhs
        typename sctl::ParallelSolver<Real>::ParallelOp BIOp = [&S,&compute_A11adj](sctl::Vector<Real>* A11adj_sigma, const sctl::Vector<Real>& sigma) {
          compute_A11adj(*A11adj_sigma, sigma);
        };
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> sigma(Nelem);
        Vector<Real> rhs_(Nelem * Nnodes), sigma_(Nelem * Nnodes);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            rhs_[i*Nnodes+j] = rhs[i][j];
            sigma_[i*Nnodes+j] = 0;
          }
        }
        ParallelSolver<Real> linear_solver(comm, true);
        linear_solver(&sigma_, BIOp, rhs_, 1e-8, 50);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            sigma[i][j] = sigma_[i*Nnodes+j];
          }
        }
        return sigma;
      };
      if (0) { // Test invA11adj
        Vector<ElemBasis> dg_dsigma = compute_dg_dsigma(B);
        Real a = compute_inner_prod(dg_dsigma, compute_invA11(sigma));
        Real b = compute_inner_prod(compute_invA11adj(dg_dsigma), sigma);
        std::cout<<a<<' '<<b<<'\n';
      }

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      // 0.168275  0.117983 -0.110446  -96.7293
      // 0.603869 -1.901900 -1.229930 -245.5050
      auto compute_u_dAdnu_v_00 = [&S,&normal,&comm,&compute_half_n_plus_dG,&compute_grad_adj] (const Vector<Real>& u_, const Vector<Real>& v_) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> u(Nelem), u_n(Nelem*COORD_DIM), v(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            u[i][j] = u_[i*Nnodes+j];
            v[i][j] = v_[i*Nnodes+j];
            u_n[i*COORD_DIM+0][j] = u[i][j] * normal[i*COORD_DIM+0][j];
            u_n[i*COORD_DIM+1][j] = u[i][j] * normal[i*COORD_DIM+1][j];
            u_n[i*COORD_DIM+2][j] = u[i][j] * normal[i*COORD_DIM+2][j];
          }
        }

        Vector<ElemBasis> dAdnu0(Nelem), dAdnu1(Nelem), dAdnu2(Nelem), dAdnu3(Nelem);
        dAdnu0 = 0;
        dAdnu1 = 0;
        dAdnu2 = 0;
        dAdnu3 = 0;

        Vector<ElemBasis> H(Nelem);
        { // Set mean curvature H
          const Vector<ElemBasis> X = S.GetElemList().ElemVector();
          Vector<ElemBasis> dX, d2X;
          ElemBasis::Grad(dX, X);
          ElemBasis::Grad(d2X, dX);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Tensor<Real,true,2,2> I, invI, II;
              for (Long k0 = 0; k0 < 2; k0++) {
                for (Long k1 = 0; k1 < 2; k1++) {
                  I(k0,k1) = 0;
                  I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                  I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                  I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                  II(k0,k1) = 0;
                  II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                  II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                  II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                }
              }
              { // Set invI
                Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                invI(0,0) = I(1,1) / detI;
                invI(0,1) = -I(0,1) / detI;
                invI(1,0) = -I(1,0) / detI;
                invI(1,1) = I(0,0) / detI;
              }
              { // Set H
                H[i][j] = 0;
                H[i][j] += -0.5 * II(0,0)*invI(0,0);
                H[i][j] += -0.5 * II(0,1)*invI(0,1);
                H[i][j] += -0.5 * II(1,0)*invI(1,0);
                H[i][j] += -0.5 * II(1,1)*invI(1,1);
              }
            }
          }
        }

        // dAdnu0 = u B \cdot grad_nu
        Vector<ElemBasis> B = compute_half_n_plus_dG(v);
        Vector<ElemBasis> u_B(Nelem*COORD_DIM);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            u_B[i*COORD_DIM+0][j] = u[i][j] * B[i*COORD_DIM+0][j];
            u_B[i*COORD_DIM+1][j] = u[i][j] * B[i*COORD_DIM+1][j];
            u_B[i*COORD_DIM+2][j] = u[i][j] * B[i*COORD_DIM+2][j];
          }
        }
        dAdnu0 = compute_grad_adj(u_B);

        // dAdnu1 = (2H) v (I/2 + \nabla G)^T [u n]
        Quadrature<Real> quadrature_dUxF;
        quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm, 0.01 * pow<-2,Real>(ORDER));
        quadrature_dUxF.Eval(dAdnu1, S.GetElemList(), u_n, S.Laplace_dUxF);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            dAdnu1[i][j] *= -2*H[i][j] * v[i][j];
          }
        }

        // dAdnu2 = (u n) \cdot (n \cdnot \nabla) \nabla G[v]
        Vector<ElemBasis> d2G_v;
        Quadrature<Real> quadrature_Fxd2U;
        quadrature_Fxd2U.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_Fxd2U, order_singular, order_direct, -1.0, comm, -0.01 * pow<-2,Real>(ORDER));
        quadrature_Fxd2U.Eval(d2G_v, S.GetElemList(), v, S.Laplace_Fxd2U);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            dAdnu2[i][j] = 0;
            dAdnu2[i][j] -= d2G_v[i*9+0][j] * normal[i*COORD_DIM+0][j] * u_n[i*COORD_DIM+0][j];
            dAdnu2[i][j] -= d2G_v[i*9+1][j] * normal[i*COORD_DIM+0][j] * u_n[i*COORD_DIM+1][j];
            dAdnu2[i][j] -= d2G_v[i*9+2][j] * normal[i*COORD_DIM+0][j] * u_n[i*COORD_DIM+2][j];

            dAdnu2[i][j] -= d2G_v[i*9+3][j] * normal[i*COORD_DIM+1][j] * u_n[i*COORD_DIM+0][j];
            dAdnu2[i][j] -= d2G_v[i*9+4][j] * normal[i*COORD_DIM+1][j] * u_n[i*COORD_DIM+1][j];
            dAdnu2[i][j] -= d2G_v[i*9+5][j] * normal[i*COORD_DIM+1][j] * u_n[i*COORD_DIM+2][j];

            dAdnu2[i][j] -= d2G_v[i*9+6][j] * normal[i*COORD_DIM+2][j] * u_n[i*COORD_DIM+0][j];
            dAdnu2[i][j] -= d2G_v[i*9+7][j] * normal[i*COORD_DIM+2][j] * u_n[i*COORD_DIM+1][j];
            dAdnu2[i][j] -= d2G_v[i*9+8][j] * normal[i*COORD_DIM+2][j] * u_n[i*COORD_DIM+2][j];
          }
        }

        // dAdnu3 = (v (\nabla D)^T [u n]
        Vector<ElemBasis> nablaDt_u_n;
        Quadrature<Real> quadrature_dUxD;
        quadrature_dUxD.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxD, order_singular, order_direct, -1.0, comm, 0.01 * pow<-2,Real>(ORDER));
        quadrature_dUxD.Eval(nablaDt_u_n, S.GetElemList(), u_n, S.Laplace_dUxD);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            dAdnu3[i][j] = 0;
            dAdnu3[i][j] += v[i][j] * nablaDt_u_n[i*COORD_DIM+0][j]*normal[i*COORD_DIM+0][j];
            dAdnu3[i][j] += v[i][j] * nablaDt_u_n[i*COORD_DIM+1][j]*normal[i*COORD_DIM+1][j];
            dAdnu3[i][j] += v[i][j] * nablaDt_u_n[i*COORD_DIM+2][j]*normal[i*COORD_DIM+2][j];
          }
        }

        return dAdnu0 + dAdnu1 + dAdnu2 + dAdnu3;
      };
      auto compute_u_dAdnu_v_01 = [&S,&comm,&compute_dB0,&normal,&area_elem,&compute_B0,&compute_grad_adj] (const Vector<Real>& u, const Vector<Real>& v) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> dAdnu(Nelem);
        Vector<ElemBasis> dB0 = compute_dB0(v[Nelem*Nnodes]);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            Real n_n_dB0 = 0;
            n_n_dB0 += dB0[i*9+0][j] * normal[i*COORD_DIM+0][j] * normal[i*COORD_DIM+0][j];
            n_n_dB0 += dB0[i*9+1][j] * normal[i*COORD_DIM+1][j] * normal[i*COORD_DIM+0][j];
            n_n_dB0 += dB0[i*9+2][j] * normal[i*COORD_DIM+2][j] * normal[i*COORD_DIM+0][j];

            n_n_dB0 += dB0[i*9+3][j] * normal[i*COORD_DIM+0][j] * normal[i*COORD_DIM+1][j];
            n_n_dB0 += dB0[i*9+4][j] * normal[i*COORD_DIM+1][j] * normal[i*COORD_DIM+1][j];
            n_n_dB0 += dB0[i*9+5][j] * normal[i*COORD_DIM+2][j] * normal[i*COORD_DIM+1][j];

            n_n_dB0 += dB0[i*9+6][j] * normal[i*COORD_DIM+0][j] * normal[i*COORD_DIM+2][j];
            n_n_dB0 += dB0[i*9+7][j] * normal[i*COORD_DIM+1][j] * normal[i*COORD_DIM+2][j];
            n_n_dB0 += dB0[i*9+8][j] * normal[i*COORD_DIM+2][j] * normal[i*COORD_DIM+2][j];

            dAdnu[i][j] = u[i*Nnodes+j] * n_n_dB0;
          }
        }

        Vector<ElemBasis> B0 = compute_B0(v[Nelem*Nnodes]);
        Vector<ElemBasis> u_B0(Nelem*COORD_DIM);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            u_B0[i*COORD_DIM+0][j] = u[i*Nnodes+j] * B0[i*COORD_DIM+0][j];
            u_B0[i*COORD_DIM+1][j] = u[i*Nnodes+j] * B0[i*COORD_DIM+1][j];
            u_B0[i*COORD_DIM+2][j] = u[i*Nnodes+j] * B0[i*COORD_DIM+2][j];
          }
        }
        dAdnu += compute_grad_adj(u_B0);

        return dAdnu;
      };
      auto compute_u_dAdnu_v_10 = [&S,&comm,&area_elem,&normal,&compute_dot_prod,&compute_grad_adj,&compute_half_n_plus_dG] (const Vector<Real>& u, const Vector<Real>& v) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> sigma(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            sigma[i][j] = v[i*Nnodes+j];
          }
        }

        auto compute_v = [&S,&area_elem] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> v(Nelem * COORD_DIM);
          Vector<ElemBasis> dX;
          ElemBasis::Grad(dX, S.GetElemList().ElemVector());
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Tensor<Real,true,COORD_DIM,2> dx;
              dx(0,0) = dX[i*COORD_DIM*2+0][j];
              dx(0,1) = dX[i*COORD_DIM*2+1][j];
              dx(1,0) = dX[i*COORD_DIM*2+2][j];
              dx(1,1) = dX[i*COORD_DIM*2+3][j];
              dx(2,0) = dX[i*COORD_DIM*2+4][j];
              dx(2,1) = dX[i*COORD_DIM*2+5][j];

              Real s = 1 / (area_elem[i][j] * S.NtNp_[0]);
              for (Long k = 0; k < COORD_DIM; k++) {
                v[i*COORD_DIM+k][j] = dx(k,1) * s;
              }
            }
          }
          return v;
        };
        auto compute_AxB = [&S] (const Vector<ElemBasis>& A, const Vector<ElemBasis>& B) {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> J(Nelem * COORD_DIM);
          for (Long i = 0; i < Nelem; i++) { // Set J
            for (Long j = 0; j < Nnodes; j++) {
              Tensor<Real,true,COORD_DIM> a, b;
              a(0) = A[i*COORD_DIM+0][j];
              a(1) = A[i*COORD_DIM+1][j];
              a(2) = A[i*COORD_DIM+2][j];

              b(0) = B[i*COORD_DIM+0][j];
              b(1) = B[i*COORD_DIM+1][j];
              b(2) = B[i*COORD_DIM+2][j];

              J[i*COORD_DIM+0][j] = a(1) * b(2) - a(2) * b(1);
              J[i*COORD_DIM+1][j] = a(2) * b(0) - a(0) * b(2);
              J[i*COORD_DIM+2][j] = a(0) * b(1) - a(1) * b(0);
            }
          }
          return J;
        };

        auto compute_dphi_dnu0 = [&S,&normal,&compute_AxB,&compute_v,&compute_half_n_plus_dG,compute_grad_adj,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);

          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> BxGv = compute_AxB(B,Gv);

          return compute_grad_adj(BxGv);
        };
        auto compute_dphi_dnu1 = [&S,&normal,&compute_AxB,&compute_v,&compute_half_n_plus_dG,&compute_dot_prod,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);

          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> BxGv = compute_AxB(B,Gv);
          Vector<ElemBasis> n_dot_BxGv = compute_dot_prod(normal,BxGv);

          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              dphi_dnu[i][j] = n_dot_BxGv[i][j] * 2*H[i][j];
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu2 = [&S,&normal,&compute_AxB,&compute_v,&compute_half_n_plus_dG,&compute_dot_prod,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> GnxB;
          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);
          S.quadrature_FxU.Eval(GnxB, S.GetElemList(), nxB, S.Laplace_FxU);

          Vector<ElemBasis> v = compute_v();
          Vector<ElemBasis> v_dot_GnxB = compute_dot_prod(v,GnxB);

          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              dphi_dnu[i][j] = v_dot_GnxB[i][j] * 2*H[i][j];
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu3 = [&S,&normal,&area_elem,&compute_AxB,&compute_half_n_plus_dG,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> GnxB;
          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);
          S.quadrature_FxU.Eval(GnxB, S.GetElemList(), nxB, S.Laplace_FxU);

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> dv_dnu1(Nelem), dv_dnu2(Nelem), dv_dnu3(Nelem);
          { // Set dv_dnu1, dv_dnu2, dv_dnu3
            Vector<ElemBasis> dX, dn, V_tmp(Nelem);
            ElemBasis::Grad(dn, normal);
            ElemBasis::Grad(dX, S.GetElemList().ElemVector());
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                dv_dnu1[i][j] = 0;
                dv_dnu1[i][j] += -GnxB[i*COORD_DIM+0][j] * dX[(i*COORD_DIM+0)*2+1][j] * 2 * H[i][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu1[i][j] += -GnxB[i*COORD_DIM+1][j] * dX[(i*COORD_DIM+1)*2+1][j] * 2 * H[i][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu1[i][j] += -GnxB[i*COORD_DIM+2][j] * dX[(i*COORD_DIM+2)*2+1][j] * 2 * H[i][j] / (area_elem[i][j] * S.NtNp_[0]);

                dv_dnu2[i][j] = 0;
                dv_dnu2[i][j] += GnxB[i*COORD_DIM+0][j] * dn[(i*COORD_DIM+0)*2+1][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu2[i][j] += GnxB[i*COORD_DIM+1][j] * dn[(i*COORD_DIM+1)*2+1][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu2[i][j] += GnxB[i*COORD_DIM+2][j] * dn[(i*COORD_DIM+2)*2+1][j] / (area_elem[i][j] * S.NtNp_[0]);

                V_tmp[i][j] = 0;
                V_tmp[i][j] += GnxB[i*COORD_DIM+0][j] * normal[i*COORD_DIM+0][j] / (area_elem[i][j] * S.NtNp_[0]); //dnu[i*2+1][j] *
                V_tmp[i][j] += GnxB[i*COORD_DIM+1][j] * normal[i*COORD_DIM+1][j] / (area_elem[i][j] * S.NtNp_[0]); //dnu[i*2+1][j] *
                V_tmp[i][j] += GnxB[i*COORD_DIM+2][j] * normal[i*COORD_DIM+2][j] / (area_elem[i][j] * S.NtNp_[0]); //dnu[i*2+1][j] *
              }
            }

            { // dv_dnu3 <-- grad_adj V_tmp
              Vector<ElemBasis> eye(Nnodes), Mgrad;
              eye = 0;
              for (Long i = 0; i < Nnodes; i++) eye[i][i] = 1;
              ElemBasis::Grad(Mgrad, eye);

              Vector<ElemBasis> grad_adj_V(Nelem);
              const auto& quad_wts = ElemBasis::QuadWts();
              for (Long i = 0; i < Nelem; i++) {
                for (Long j = 0; j < Nnodes; j++) {
                  Real sum = 0;
                  for (Long k = 0; k < Nnodes; k++) {
                    sum += Mgrad[j*2+1][k] * V_tmp[i][k] * (area_elem[i][k] * quad_wts[k]) / (quad_wts[j] * area_elem[i][j]);
                  }
                  dv_dnu3[i][j] = sum;
                }
              }
            }
          }

          return dv_dnu1+dv_dnu2+dv_dnu3;
        };
        auto compute_dphi_dnu4 = [&S,&normal,&compute_AxB,&compute_v,&compute_half_n_plus_dG,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> dGnxB;
          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);
          S.quadrature_FxdU.Eval(dGnxB, S.GetElemList(), nxB, S.Laplace_FxdU);

          Vector<ElemBasis> v = compute_v();
          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Real dphi_dnu_ = 0;
              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGnxB[(i*COORD_DIM+0)*COORD_DIM+0][j] * v[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGnxB[(i*COORD_DIM+0)*COORD_DIM+1][j] * v[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGnxB[(i*COORD_DIM+0)*COORD_DIM+2][j] * v[i*COORD_DIM+0][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGnxB[(i*COORD_DIM+1)*COORD_DIM+0][j] * v[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGnxB[(i*COORD_DIM+1)*COORD_DIM+1][j] * v[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGnxB[(i*COORD_DIM+1)*COORD_DIM+2][j] * v[i*COORD_DIM+1][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGnxB[(i*COORD_DIM+2)*COORD_DIM+0][j] * v[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGnxB[(i*COORD_DIM+2)*COORD_DIM+1][j] * v[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGnxB[(i*COORD_DIM+2)*COORD_DIM+2][j] * v[i*COORD_DIM+2][j];
              dphi_dnu[i][j] = dphi_dnu_;
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu5 = [&S,&normal,&compute_AxB,&compute_v,&compute_half_n_plus_dG,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);

          Vector<ElemBasis> dGv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxdU.Eval(dGv, S.GetElemList(), v, S.Laplace_FxdU);

          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Real dphi_dnu_ = 0;
              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGv[(i*COORD_DIM+0)*COORD_DIM+0][j] * nxB[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGv[(i*COORD_DIM+0)*COORD_DIM+1][j] * nxB[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGv[(i*COORD_DIM+0)*COORD_DIM+2][j] * nxB[i*COORD_DIM+0][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGv[(i*COORD_DIM+1)*COORD_DIM+0][j] * nxB[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGv[(i*COORD_DIM+1)*COORD_DIM+1][j] * nxB[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGv[(i*COORD_DIM+1)*COORD_DIM+2][j] * nxB[i*COORD_DIM+1][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGv[(i*COORD_DIM+2)*COORD_DIM+0][j] * nxB[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGv[(i*COORD_DIM+2)*COORD_DIM+1][j] * nxB[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGv[(i*COORD_DIM+2)*COORD_DIM+2][j] * nxB[i*COORD_DIM+2][j];
              dphi_dnu[i][j] = dphi_dnu_;
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu6 = [&S,&normal,&compute_AxB,&compute_v,sigma,&comm] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);
          Vector<ElemBasis> nxGv = compute_AxB(Gv,normal);

          Vector<ElemBasis> gradB;
          Quadrature<Real> quadrature_Fxd2U;
          quadrature_Fxd2U.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_Fxd2U, order_singular, order_direct, -1.0, comm, -0.01 * pow<-2,Real>(ORDER));
          quadrature_Fxd2U.Eval(gradB, S.GetElemList(), sigma, S.Laplace_Fxd2U);
          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Real dphi_dnu_ = 0;
              dphi_dnu_ += -nxGv[i*COORD_DIM+0][j] * gradB[(i*COORD_DIM+0)*COORD_DIM+0][j] * normal[i*COORD_DIM+0][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+1][j] * gradB[(i*COORD_DIM+0)*COORD_DIM+1][j] * normal[i*COORD_DIM+0][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+2][j] * gradB[(i*COORD_DIM+0)*COORD_DIM+2][j] * normal[i*COORD_DIM+0][j];

              dphi_dnu_ += -nxGv[i*COORD_DIM+0][j] * gradB[(i*COORD_DIM+1)*COORD_DIM+0][j] * normal[i*COORD_DIM+1][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+1][j] * gradB[(i*COORD_DIM+1)*COORD_DIM+1][j] * normal[i*COORD_DIM+1][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+2][j] * gradB[(i*COORD_DIM+1)*COORD_DIM+2][j] * normal[i*COORD_DIM+1][j];

              dphi_dnu_ += -nxGv[i*COORD_DIM+0][j] * gradB[(i*COORD_DIM+2)*COORD_DIM+0][j] * normal[i*COORD_DIM+2][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+1][j] * gradB[(i*COORD_DIM+2)*COORD_DIM+1][j] * normal[i*COORD_DIM+2][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+2][j] * gradB[(i*COORD_DIM+2)*COORD_DIM+2][j] * normal[i*COORD_DIM+2][j];
              dphi_dnu[i][j] = dphi_dnu_;
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu7 = [&S,&normal,&compute_AxB,&compute_v,sigma,&comm] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);
          Vector<ElemBasis> nxGv = compute_AxB(Gv,normal);

          Vector<ElemBasis> dphi_dnu(Nelem);
          Quadrature<Real> quadrature_dUxF;
          quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm, 0.01 * pow<-2,Real>(ORDER));
          quadrature_dUxF.Eval(dphi_dnu, S.GetElemList(), nxGv, S.Laplace_dUxF);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              dphi_dnu[i][j] *= -2*H[i][j] * sigma[i][j];
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu8 = [&S,&normal,&compute_AxB,&compute_v,sigma,&comm] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);
          Vector<ElemBasis> nxGv = compute_AxB(Gv,normal);

          Vector<ElemBasis> dphi_dnu(Nelem);
          Vector<ElemBasis> nablaDt_nxGv;
          Quadrature<Real> quadrature_dUxD;
          quadrature_dUxD.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxD, order_singular, order_direct, -1.0, comm, 0.01 * pow<-2,Real>(ORDER));
          quadrature_dUxD.Eval(nablaDt_nxGv, S.GetElemList(), nxGv, S.Laplace_dUxD);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              dphi_dnu[i][j] = 0;
              dphi_dnu[i][j] += sigma[i][j] * nablaDt_nxGv[i*COORD_DIM+0][j]*normal[i*COORD_DIM+0][j];
              dphi_dnu[i][j] += sigma[i][j] * nablaDt_nxGv[i*COORD_DIM+1][j]*normal[i*COORD_DIM+1][j];
              dphi_dnu[i][j] += sigma[i][j] * nablaDt_nxGv[i*COORD_DIM+2][j]*normal[i*COORD_DIM+2][j];
            }
          }
          return dphi_dnu;
        };

        auto dphi_dnu0 = compute_dphi_dnu0();
        auto dphi_dnu1 = compute_dphi_dnu1();
        auto dphi_dnu2 = compute_dphi_dnu2();
        auto dphi_dnu3 = compute_dphi_dnu3();
        auto dphi_dnu4 = compute_dphi_dnu4();
        auto dphi_dnu5 = compute_dphi_dnu5();
        auto dphi_dnu6 = compute_dphi_dnu6();
        auto dphi_dnu7 = compute_dphi_dnu7();
        auto dphi_dnu8 = compute_dphi_dnu8();

        return (dphi_dnu0+dphi_dnu1+dphi_dnu2+dphi_dnu3+dphi_dnu4+dphi_dnu5+dphi_dnu6+dphi_dnu7+dphi_dnu8) * u[Nelem*Nnodes];
      };
      auto compute_u_dAdnu_v_11 = [&S,&comm,&area_elem,&normal,&compute_dot_prod,&compute_grad_adj,&compute_B0,&compute_dB0] (const Vector<Real>& u, const Vector<Real>& v) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        auto compute_v = [&S,&area_elem] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> v(Nelem * COORD_DIM);
          Vector<ElemBasis> dX;
          ElemBasis::Grad(dX, S.GetElemList().ElemVector());
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Tensor<Real,true,COORD_DIM,2> dx;
              dx(0,0) = dX[i*COORD_DIM*2+0][j];
              dx(0,1) = dX[i*COORD_DIM*2+1][j];
              dx(1,0) = dX[i*COORD_DIM*2+2][j];
              dx(1,1) = dX[i*COORD_DIM*2+3][j];
              dx(2,0) = dX[i*COORD_DIM*2+4][j];
              dx(2,1) = dX[i*COORD_DIM*2+5][j];

              Real s = 1 / (area_elem[i][j] * S.NtNp_[0]);
              for (Long k = 0; k < COORD_DIM; k++) {
                v[i*COORD_DIM+k][j] = dx(k,1) * s;
              }
            }
          }
          return v;
        };
        auto compute_AxB = [&S] (const Vector<ElemBasis>& A, const Vector<ElemBasis>& B) {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> J(Nelem * COORD_DIM);
          for (Long i = 0; i < Nelem; i++) { // Set J
            for (Long j = 0; j < Nnodes; j++) {
              Tensor<Real,true,COORD_DIM> a, b;
              a(0) = A[i*COORD_DIM+0][j];
              a(1) = A[i*COORD_DIM+1][j];
              a(2) = A[i*COORD_DIM+2][j];

              b(0) = B[i*COORD_DIM+0][j];
              b(1) = B[i*COORD_DIM+1][j];
              b(2) = B[i*COORD_DIM+2][j];

              J[i*COORD_DIM+0][j] = a(1) * b(2) - a(2) * b(1);
              J[i*COORD_DIM+1][j] = a(2) * b(0) - a(0) * b(2);
              J[i*COORD_DIM+2][j] = a(0) * b(1) - a(1) * b(0);
            }
          }
          return J;
        };

        auto compute_dphi_dnu0 = [&S,&normal,&compute_AxB,&compute_v,&compute_dB0] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);
          Vector<ElemBasis> nxGv = compute_AxB(Gv,normal);

          Vector<ElemBasis> gradB = compute_dB0(1.0);
          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Real dphi_dnu_ = 0;
              dphi_dnu_ += nxGv[i*COORD_DIM+0][j] * gradB[(i*COORD_DIM+0)*COORD_DIM+0][j] * normal[i*COORD_DIM+0][j];
              dphi_dnu_ += nxGv[i*COORD_DIM+1][j] * gradB[(i*COORD_DIM+0)*COORD_DIM+1][j] * normal[i*COORD_DIM+0][j];
              dphi_dnu_ += nxGv[i*COORD_DIM+2][j] * gradB[(i*COORD_DIM+0)*COORD_DIM+2][j] * normal[i*COORD_DIM+0][j];

              dphi_dnu_ += nxGv[i*COORD_DIM+0][j] * gradB[(i*COORD_DIM+1)*COORD_DIM+0][j] * normal[i*COORD_DIM+1][j];
              dphi_dnu_ += nxGv[i*COORD_DIM+1][j] * gradB[(i*COORD_DIM+1)*COORD_DIM+1][j] * normal[i*COORD_DIM+1][j];
              dphi_dnu_ += nxGv[i*COORD_DIM+2][j] * gradB[(i*COORD_DIM+1)*COORD_DIM+2][j] * normal[i*COORD_DIM+1][j];

              dphi_dnu_ += nxGv[i*COORD_DIM+0][j] * gradB[(i*COORD_DIM+2)*COORD_DIM+0][j] * normal[i*COORD_DIM+2][j];
              dphi_dnu_ += nxGv[i*COORD_DIM+1][j] * gradB[(i*COORD_DIM+2)*COORD_DIM+1][j] * normal[i*COORD_DIM+2][j];
              dphi_dnu_ += nxGv[i*COORD_DIM+2][j] * gradB[(i*COORD_DIM+2)*COORD_DIM+2][j] * normal[i*COORD_DIM+2][j];
              dphi_dnu[i][j] = dphi_dnu_;
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu1 = [&S,&normal,&compute_AxB,&compute_v,&compute_B0,compute_grad_adj] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);

          Vector<ElemBasis> B = compute_B0(1.0);
          Vector<ElemBasis> BxGv = compute_AxB(B,Gv);

          return compute_grad_adj(BxGv);
        };
        auto compute_dphi_dnu2 = [&S,&normal,&compute_AxB,&compute_v,&compute_B0,&compute_dot_prod] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);

          Vector<ElemBasis> B = compute_B0(1.0);
          Vector<ElemBasis> BxGv = compute_AxB(B,Gv);
          Vector<ElemBasis> n_dot_BxGv = compute_dot_prod(normal,BxGv);

          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              dphi_dnu[i][j] = n_dot_BxGv[i][j] * 2*H[i][j];
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu3 = [&S,&normal,&compute_AxB,&compute_v,&compute_B0,&compute_dot_prod] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> GnxB;
          Vector<ElemBasis> B = compute_B0(1.0);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);
          S.quadrature_FxU.Eval(GnxB, S.GetElemList(), nxB, S.Laplace_FxU);

          Vector<ElemBasis> v = compute_v();
          Vector<ElemBasis> v_dot_GnxB = compute_dot_prod(v,GnxB);

          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              dphi_dnu[i][j] = v_dot_GnxB[i][j] * 2*H[i][j];
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu4 = [&S,&normal,&area_elem,&compute_AxB,&compute_B0] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> GnxB;
          Vector<ElemBasis> B = compute_B0(1.0);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);
          S.quadrature_FxU.Eval(GnxB, S.GetElemList(), nxB, S.Laplace_FxU);

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> dv_dnu1(Nelem), dv_dnu2(Nelem), dv_dnu3(Nelem);
          { // Set dv_dnu1, dv_dnu2, dv_dnu3
            Vector<ElemBasis> dX, dn, V_tmp(Nelem);
            ElemBasis::Grad(dn, normal);
            ElemBasis::Grad(dX, S.GetElemList().ElemVector());
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                dv_dnu1[i][j] = 0;
                dv_dnu1[i][j] += -GnxB[i*COORD_DIM+0][j] * dX[(i*COORD_DIM+0)*2+1][j] * 2 * H[i][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu1[i][j] += -GnxB[i*COORD_DIM+1][j] * dX[(i*COORD_DIM+1)*2+1][j] * 2 * H[i][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu1[i][j] += -GnxB[i*COORD_DIM+2][j] * dX[(i*COORD_DIM+2)*2+1][j] * 2 * H[i][j] / (area_elem[i][j] * S.NtNp_[0]);

                dv_dnu2[i][j] = 0;
                dv_dnu2[i][j] += GnxB[i*COORD_DIM+0][j] * dn[(i*COORD_DIM+0)*2+1][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu2[i][j] += GnxB[i*COORD_DIM+1][j] * dn[(i*COORD_DIM+1)*2+1][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu2[i][j] += GnxB[i*COORD_DIM+2][j] * dn[(i*COORD_DIM+2)*2+1][j] / (area_elem[i][j] * S.NtNp_[0]);

                V_tmp[i][j] = 0;
                V_tmp[i][j] += GnxB[i*COORD_DIM+0][j] * normal[i*COORD_DIM+0][j] / (area_elem[i][j] * S.NtNp_[0]); //dnu[i*2+1][j] *
                V_tmp[i][j] += GnxB[i*COORD_DIM+1][j] * normal[i*COORD_DIM+1][j] / (area_elem[i][j] * S.NtNp_[0]); //dnu[i*2+1][j] *
                V_tmp[i][j] += GnxB[i*COORD_DIM+2][j] * normal[i*COORD_DIM+2][j] / (area_elem[i][j] * S.NtNp_[0]); //dnu[i*2+1][j] *
              }
            }

            { // dv_dnu3 <-- grad_adj V_tmp
              Vector<ElemBasis> eye(Nnodes), Mgrad;
              eye = 0;
              for (Long i = 0; i < Nnodes; i++) eye[i][i] = 1;
              ElemBasis::Grad(Mgrad, eye);

              Vector<ElemBasis> grad_adj_V(Nelem);
              const auto& quad_wts = ElemBasis::QuadWts();
              for (Long i = 0; i < Nelem; i++) {
                for (Long j = 0; j < Nnodes; j++) {
                  Real sum = 0;
                  for (Long k = 0; k < Nnodes; k++) {
                    sum += Mgrad[j*2+1][k] * V_tmp[i][k] * (area_elem[i][k] * quad_wts[k]) / (quad_wts[j] * area_elem[i][j]);
                  }
                  dv_dnu3[i][j] = sum;
                }
              }
            }
          }

          return dv_dnu1+dv_dnu2+dv_dnu3;
        };
        auto compute_dphi_dnu5 = [&S,&normal,&compute_AxB,&compute_v,&compute_B0] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> dGnxB;
          Vector<ElemBasis> B = compute_B0(1.0);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);
          S.quadrature_FxdU.Eval(dGnxB, S.GetElemList(), nxB, S.Laplace_FxdU);

          Vector<ElemBasis> v = compute_v();
          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Real dphi_dnu_ = 0;
              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGnxB[(i*COORD_DIM+0)*COORD_DIM+0][j] * v[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGnxB[(i*COORD_DIM+0)*COORD_DIM+1][j] * v[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGnxB[(i*COORD_DIM+0)*COORD_DIM+2][j] * v[i*COORD_DIM+0][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGnxB[(i*COORD_DIM+1)*COORD_DIM+0][j] * v[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGnxB[(i*COORD_DIM+1)*COORD_DIM+1][j] * v[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGnxB[(i*COORD_DIM+1)*COORD_DIM+2][j] * v[i*COORD_DIM+1][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGnxB[(i*COORD_DIM+2)*COORD_DIM+0][j] * v[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGnxB[(i*COORD_DIM+2)*COORD_DIM+1][j] * v[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGnxB[(i*COORD_DIM+2)*COORD_DIM+2][j] * v[i*COORD_DIM+2][j];
              dphi_dnu[i][j] = dphi_dnu_;
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu6 = [&S,&normal,&compute_AxB,&compute_v,&compute_B0] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> B = compute_B0(1.0);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);

          Vector<ElemBasis> dGv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxdU.Eval(dGv, S.GetElemList(), v, S.Laplace_FxdU);

          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Real dphi_dnu_ = 0;
              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGv[(i*COORD_DIM+0)*COORD_DIM+0][j] * nxB[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGv[(i*COORD_DIM+0)*COORD_DIM+1][j] * nxB[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGv[(i*COORD_DIM+0)*COORD_DIM+2][j] * nxB[i*COORD_DIM+0][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGv[(i*COORD_DIM+1)*COORD_DIM+0][j] * nxB[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGv[(i*COORD_DIM+1)*COORD_DIM+1][j] * nxB[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGv[(i*COORD_DIM+1)*COORD_DIM+2][j] * nxB[i*COORD_DIM+1][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGv[(i*COORD_DIM+2)*COORD_DIM+0][j] * nxB[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGv[(i*COORD_DIM+2)*COORD_DIM+1][j] * nxB[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGv[(i*COORD_DIM+2)*COORD_DIM+2][j] * nxB[i*COORD_DIM+2][j];
              dphi_dnu[i][j] = dphi_dnu_;
            }
          }
          return dphi_dnu;
        };

        auto dphi_dnu0 = compute_dphi_dnu0();
        auto dphi_dnu1 = compute_dphi_dnu1();
        auto dphi_dnu2 = compute_dphi_dnu2();
        auto dphi_dnu3 = compute_dphi_dnu3();
        auto dphi_dnu4 = compute_dphi_dnu4();
        auto dphi_dnu5 = compute_dphi_dnu5();
        auto dphi_dnu6 = compute_dphi_dnu6();

        return (dphi_dnu0+dphi_dnu1+dphi_dnu2+dphi_dnu3+dphi_dnu4+dphi_dnu5+dphi_dnu6) * (u[Nelem*Nnodes] * v[Nelem*Nnodes]);
      };

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      if (0) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        auto compute_v = [&S,&area_elem] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> v(Nelem * COORD_DIM);
          Vector<ElemBasis> dX;
          ElemBasis::Grad(dX, S.GetElemList().ElemVector());
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Tensor<Real,true,COORD_DIM,2> dx;
              dx(0,0) = dX[i*COORD_DIM*2+0][j];
              dx(0,1) = dX[i*COORD_DIM*2+1][j];
              dx(1,0) = dX[i*COORD_DIM*2+2][j];
              dx(1,1) = dX[i*COORD_DIM*2+3][j];
              dx(2,0) = dX[i*COORD_DIM*2+4][j];
              dx(2,1) = dX[i*COORD_DIM*2+5][j];

              Real s = 1 / (area_elem[i][j] * S.NtNp_[0]);
              for (Long k = 0; k < COORD_DIM; k++) {
                v[i*COORD_DIM+k][j] = dx(k,1) * s;
              }
            }
          }
          return v;
        };
        auto compute_AxB = [&S] (const Vector<ElemBasis>& A, const Vector<ElemBasis>& B) {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> J(Nelem * COORD_DIM);
          for (Long i = 0; i < Nelem; i++) { // Set J
            for (Long j = 0; j < Nnodes; j++) {
              Tensor<Real,true,COORD_DIM> a, b;
              a(0) = A[i*COORD_DIM+0][j];
              a(1) = A[i*COORD_DIM+1][j];
              a(2) = A[i*COORD_DIM+2][j];

              b(0) = B[i*COORD_DIM+0][j];
              b(1) = B[i*COORD_DIM+1][j];
              b(2) = B[i*COORD_DIM+2][j];

              J[i*COORD_DIM+0][j] = a(1) * b(2) - a(2) * b(1);
              J[i*COORD_DIM+1][j] = a(2) * b(0) - a(0) * b(2);
              J[i*COORD_DIM+2][j] = a(0) * b(1) - a(1) * b(0);
            }
          }
          return J;
        };

        auto compute_dphi_dnu0 = [&S,&normal,&compute_AxB,&compute_v,&compute_half_n_plus_dG,compute_grad_adj,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);

          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> BxGv = compute_AxB(B,Gv);

          return compute_grad_adj(BxGv);
        };
        auto compute_dphi_dnu1 = [&S,&normal,&compute_AxB,&compute_v,&compute_half_n_plus_dG,&compute_dot_prod,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);

          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> BxGv = compute_AxB(B,Gv);
          Vector<ElemBasis> n_dot_BxGv = compute_dot_prod(normal,BxGv);

          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              dphi_dnu[i][j] = n_dot_BxGv[i][j] * 2*H[i][j];
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu2 = [&S,&normal,&compute_AxB,&compute_v,&compute_half_n_plus_dG,&compute_dot_prod,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> GnxB;
          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);
          S.quadrature_FxU.Eval(GnxB, S.GetElemList(), nxB, S.Laplace_FxU);

          Vector<ElemBasis> v = compute_v();
          Vector<ElemBasis> v_dot_GnxB = compute_dot_prod(v,GnxB);

          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              dphi_dnu[i][j] = v_dot_GnxB[i][j] * 2*H[i][j];
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu3 = [&S,&normal,&area_elem,&compute_AxB,&compute_half_n_plus_dG,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> GnxB;
          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);
          S.quadrature_FxU.Eval(GnxB, S.GetElemList(), nxB, S.Laplace_FxU);

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> dv_dnu1(Nelem), dv_dnu2(Nelem), dv_dnu3(Nelem);
          { // Set dv_dnu1, dv_dnu2, dv_dnu3
            Vector<ElemBasis> dX, dn, V_tmp(Nelem);
            ElemBasis::Grad(dn, normal);
            ElemBasis::Grad(dX, S.GetElemList().ElemVector());
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                dv_dnu1[i][j] = 0;
                dv_dnu1[i][j] += -GnxB[i*COORD_DIM+0][j] * dX[(i*COORD_DIM+0)*2+1][j] * 2 * H[i][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu1[i][j] += -GnxB[i*COORD_DIM+1][j] * dX[(i*COORD_DIM+1)*2+1][j] * 2 * H[i][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu1[i][j] += -GnxB[i*COORD_DIM+2][j] * dX[(i*COORD_DIM+2)*2+1][j] * 2 * H[i][j] / (area_elem[i][j] * S.NtNp_[0]);

                dv_dnu2[i][j] = 0;
                dv_dnu2[i][j] += GnxB[i*COORD_DIM+0][j] * dn[(i*COORD_DIM+0)*2+1][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu2[i][j] += GnxB[i*COORD_DIM+1][j] * dn[(i*COORD_DIM+1)*2+1][j] / (area_elem[i][j] * S.NtNp_[0]);
                dv_dnu2[i][j] += GnxB[i*COORD_DIM+2][j] * dn[(i*COORD_DIM+2)*2+1][j] / (area_elem[i][j] * S.NtNp_[0]);

                V_tmp[i][j] = 0;
                V_tmp[i][j] += GnxB[i*COORD_DIM+0][j] * normal[i*COORD_DIM+0][j] / (area_elem[i][j] * S.NtNp_[0]); //dnu[i*2+1][j] *
                V_tmp[i][j] += GnxB[i*COORD_DIM+1][j] * normal[i*COORD_DIM+1][j] / (area_elem[i][j] * S.NtNp_[0]); //dnu[i*2+1][j] *
                V_tmp[i][j] += GnxB[i*COORD_DIM+2][j] * normal[i*COORD_DIM+2][j] / (area_elem[i][j] * S.NtNp_[0]); //dnu[i*2+1][j] *
              }
            }

            { // dv_dnu3 <-- grad_adj V_tmp
              Vector<ElemBasis> eye(Nnodes), Mgrad;
              eye = 0;
              for (Long i = 0; i < Nnodes; i++) eye[i][i] = 1;
              ElemBasis::Grad(Mgrad, eye);

              Vector<ElemBasis> grad_adj_V(Nelem);
              const auto& quad_wts = ElemBasis::QuadWts();
              for (Long i = 0; i < Nelem; i++) {
                for (Long j = 0; j < Nnodes; j++) {
                  Real sum = 0;
                  for (Long k = 0; k < Nnodes; k++) {
                    sum += Mgrad[j*2+1][k] * V_tmp[i][k] * (area_elem[i][k] * quad_wts[k]) / (quad_wts[j] * area_elem[i][j]);
                  }
                  dv_dnu3[i][j] = sum;
                }
              }
            }
          }

          return dv_dnu1+dv_dnu2+dv_dnu3;
        };
        auto compute_dphi_dnu4 = [&S,&normal,&compute_AxB,&compute_v,&compute_half_n_plus_dG,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> dGnxB;
          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);
          S.quadrature_FxdU.Eval(dGnxB, S.GetElemList(), nxB, S.Laplace_FxdU);

          Vector<ElemBasis> v = compute_v();
          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Real dphi_dnu_ = 0;
              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGnxB[(i*COORD_DIM+0)*COORD_DIM+0][j] * v[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGnxB[(i*COORD_DIM+0)*COORD_DIM+1][j] * v[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGnxB[(i*COORD_DIM+0)*COORD_DIM+2][j] * v[i*COORD_DIM+0][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGnxB[(i*COORD_DIM+1)*COORD_DIM+0][j] * v[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGnxB[(i*COORD_DIM+1)*COORD_DIM+1][j] * v[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGnxB[(i*COORD_DIM+1)*COORD_DIM+2][j] * v[i*COORD_DIM+1][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGnxB[(i*COORD_DIM+2)*COORD_DIM+0][j] * v[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGnxB[(i*COORD_DIM+2)*COORD_DIM+1][j] * v[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGnxB[(i*COORD_DIM+2)*COORD_DIM+2][j] * v[i*COORD_DIM+2][j];
              dphi_dnu[i][j] = dphi_dnu_;
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu5 = [&S,&normal,&compute_AxB,&compute_v,&compute_half_n_plus_dG,sigma] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> nxB = compute_AxB(normal,B);

          Vector<ElemBasis> dGv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxdU.Eval(dGv, S.GetElemList(), v, S.Laplace_FxdU);

          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Real dphi_dnu_ = 0;
              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGv[(i*COORD_DIM+0)*COORD_DIM+0][j] * nxB[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGv[(i*COORD_DIM+0)*COORD_DIM+1][j] * nxB[i*COORD_DIM+0][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGv[(i*COORD_DIM+0)*COORD_DIM+2][j] * nxB[i*COORD_DIM+0][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGv[(i*COORD_DIM+1)*COORD_DIM+0][j] * nxB[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGv[(i*COORD_DIM+1)*COORD_DIM+1][j] * nxB[i*COORD_DIM+1][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGv[(i*COORD_DIM+1)*COORD_DIM+2][j] * nxB[i*COORD_DIM+1][j];

              dphi_dnu_ += -normal[i*COORD_DIM+0][j] * dGv[(i*COORD_DIM+2)*COORD_DIM+0][j] * nxB[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+1][j] * dGv[(i*COORD_DIM+2)*COORD_DIM+1][j] * nxB[i*COORD_DIM+2][j];
              dphi_dnu_ += -normal[i*COORD_DIM+2][j] * dGv[(i*COORD_DIM+2)*COORD_DIM+2][j] * nxB[i*COORD_DIM+2][j];
              dphi_dnu[i][j] = dphi_dnu_;
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu6 = [&S,&normal,&compute_AxB,&compute_v,sigma,&comm] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);
          Vector<ElemBasis> nxGv = compute_AxB(Gv,normal);

          Vector<ElemBasis> gradB;
          Quadrature<Real> quadrature_Fxd2U;
          quadrature_Fxd2U.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_Fxd2U, order_singular, order_direct, -1.0, comm, -0.01 * pow<-2,Real>(ORDER));
          quadrature_Fxd2U.Eval(gradB, S.GetElemList(), sigma, S.Laplace_Fxd2U);
          Vector<ElemBasis> dphi_dnu(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              Real dphi_dnu_ = 0;
              dphi_dnu_ += -nxGv[i*COORD_DIM+0][j] * gradB[(i*COORD_DIM+0)*COORD_DIM+0][j] * normal[i*COORD_DIM+0][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+1][j] * gradB[(i*COORD_DIM+0)*COORD_DIM+1][j] * normal[i*COORD_DIM+0][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+2][j] * gradB[(i*COORD_DIM+0)*COORD_DIM+2][j] * normal[i*COORD_DIM+0][j];

              dphi_dnu_ += -nxGv[i*COORD_DIM+0][j] * gradB[(i*COORD_DIM+1)*COORD_DIM+0][j] * normal[i*COORD_DIM+1][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+1][j] * gradB[(i*COORD_DIM+1)*COORD_DIM+1][j] * normal[i*COORD_DIM+1][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+2][j] * gradB[(i*COORD_DIM+1)*COORD_DIM+2][j] * normal[i*COORD_DIM+1][j];

              dphi_dnu_ += -nxGv[i*COORD_DIM+0][j] * gradB[(i*COORD_DIM+2)*COORD_DIM+0][j] * normal[i*COORD_DIM+2][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+1][j] * gradB[(i*COORD_DIM+2)*COORD_DIM+1][j] * normal[i*COORD_DIM+2][j];
              dphi_dnu_ += -nxGv[i*COORD_DIM+2][j] * gradB[(i*COORD_DIM+2)*COORD_DIM+2][j] * normal[i*COORD_DIM+2][j];
              dphi_dnu[i][j] = dphi_dnu_;
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu7 = [&S,&normal,&compute_AxB,&compute_v,sigma,&comm] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);
          Vector<ElemBasis> nxGv = compute_AxB(Gv,normal);

          Vector<ElemBasis> dphi_dnu(Nelem);
          Quadrature<Real> quadrature_dUxF;
          quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm, 0.01 * pow<-2,Real>(ORDER));
          quadrature_dUxF.Eval(dphi_dnu, S.GetElemList(), nxGv, S.Laplace_dUxF);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              dphi_dnu[i][j] *= -2*H[i][j] * sigma[i][j];
            }
          }
          return dphi_dnu;
        };
        auto compute_dphi_dnu8 = [&S,&normal,&compute_AxB,&compute_v,sigma,&comm] () {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> H(Nelem);
          { // Set mean curvature H
            const Vector<ElemBasis> X = S.GetElemList().ElemVector();
            Vector<ElemBasis> dX, d2X;
            ElemBasis::Grad(dX, X);
            ElemBasis::Grad(d2X, dX);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                Tensor<Real,true,2,2> I, invI, II;
                for (Long k0 = 0; k0 < 2; k0++) {
                  for (Long k1 = 0; k1 < 2; k1++) {
                    I(k0,k1) = 0;
                    I(k0,k1) += dX[(i*COORD_DIM+0)*2+k0][j] * dX[(i*COORD_DIM+0)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+1)*2+k0][j] * dX[(i*COORD_DIM+1)*2+k1][j];
                    I(k0,k1) += dX[(i*COORD_DIM+2)*2+k0][j] * dX[(i*COORD_DIM+2)*2+k1][j];

                    II(k0,k1) = 0;
                    II(k0,k1) += d2X[(i*COORD_DIM+0)*4+k0*2+k1][j] * normal[i*COORD_DIM+0][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+1)*4+k0*2+k1][j] * normal[i*COORD_DIM+1][j];
                    II(k0,k1) += d2X[(i*COORD_DIM+2)*4+k0*2+k1][j] * normal[i*COORD_DIM+2][j];
                  }
                }
                { // Set invI
                  Real detI = I(0,0)*I(1,1)-I(0,1)*I(1,0);
                  invI(0,0) = I(1,1) / detI;
                  invI(0,1) = -I(0,1) / detI;
                  invI(1,0) = -I(1,0) / detI;
                  invI(1,1) = I(0,0) / detI;
                }
                { // Set H
                  H[i][j] = 0;
                  H[i][j] += -0.5 * II(0,0)*invI(0,0);
                  H[i][j] += -0.5 * II(0,1)*invI(0,1);
                  H[i][j] += -0.5 * II(1,0)*invI(1,0);
                  H[i][j] += -0.5 * II(1,1)*invI(1,1);
                }
              }
            }
          }

          Vector<ElemBasis> Gv;
          Vector<ElemBasis> v = compute_v();
          S.quadrature_FxU.Eval(Gv, S.GetElemList(), v, S.Laplace_FxU);
          Vector<ElemBasis> nxGv = compute_AxB(Gv,normal);

          Vector<ElemBasis> dphi_dnu(Nelem);
          Vector<ElemBasis> nablaDt_nxGv;
          Quadrature<Real> quadrature_dUxD;
          quadrature_dUxD.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxD, order_singular, order_direct, -1.0, comm, 0.01 * pow<-2,Real>(ORDER));
          quadrature_dUxD.Eval(nablaDt_nxGv, S.GetElemList(), nxGv, S.Laplace_dUxD);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              dphi_dnu[i][j] = 0;
              dphi_dnu[i][j] += sigma[i][j] * nablaDt_nxGv[i*COORD_DIM+0][j]*normal[i*COORD_DIM+0][j];
              dphi_dnu[i][j] += sigma[i][j] * nablaDt_nxGv[i*COORD_DIM+1][j]*normal[i*COORD_DIM+1][j];
              dphi_dnu[i][j] += sigma[i][j] * nablaDt_nxGv[i*COORD_DIM+2][j]*normal[i*COORD_DIM+2][j];
            }
          }
          return dphi_dnu;
        };

        Vector<ElemBasis> nu(Nelem);
        nu = 1; //area_elem;

        Real dphi_dnu0 = compute_inner_prod(nu, compute_dphi_dnu0());
        Real dphi_dnu1 = compute_inner_prod(nu, compute_dphi_dnu1());
        Real dphi_dnu2 = compute_inner_prod(nu, compute_dphi_dnu2());
        Real dphi_dnu3 = compute_inner_prod(nu, compute_dphi_dnu3());
        Real dphi_dnu4 = compute_inner_prod(nu, compute_dphi_dnu4());
        Real dphi_dnu5 = compute_inner_prod(nu, compute_dphi_dnu5());
        Real dphi_dnu6 = compute_inner_prod(nu, compute_dphi_dnu6());
        Real dphi_dnu7 = compute_inner_prod(nu, compute_dphi_dnu7());
        Real dphi_dnu8 = compute_inner_prod(nu, compute_dphi_dnu8());
        std::cout<<dphi_dnu0<<' ';
        std::cout<<dphi_dnu1<<' ';
        std::cout<<dphi_dnu2<<' ';
        std::cout<<dphi_dnu3<<' ';
        std::cout<<dphi_dnu4<<' ';
        std::cout<<dphi_dnu5<<' ';
        std::cout<<dphi_dnu6<<' ';
        std::cout<<dphi_dnu7<<' ';
        std::cout<<dphi_dnu8<<' ';
        std::cout<<'\n';
        std::cout<<dphi_dnu0+dphi_dnu1+dphi_dnu2+dphi_dnu3+dphi_dnu4+dphi_dnu5+dphi_dnu6+dphi_dnu7+dphi_dnu8<<'\n';

        auto compute_flux = [&S,&comm,&normal,&area_elem,&compute_norm_area_elem,&compute_AxB,&compute_v,&compute_inner_prod,&sigma,&compute_half_n_plus_dG] (const Vector<ElemBasis>& nu, Real eps) {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> X_orig(Nelem*COORD_DIM);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              X_orig[i*COORD_DIM+0][j] = S.Elem(i,0)[j];
              X_orig[i*COORD_DIM+1][j] = S.Elem(i,1)[j];
              X_orig[i*COORD_DIM+2][j] = S.Elem(i,2)[j];
              S.Elem(i,0)[j] += eps*nu[i][j] * normal[i*COORD_DIM+0][j];
              S.Elem(i,1)[j] += eps*nu[i][j] * normal[i*COORD_DIM+1][j];
              S.Elem(i,2)[j] += eps*nu[i][j] * normal[i*COORD_DIM+2][j];
            }
          }
          compute_norm_area_elem(normal, area_elem);
          S.quadrature_FxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxU , order_singular, order_direct, -1.0, comm);
          S.quadrature_FxdU.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxdU, order_singular, order_direct, -1.0, comm);

          Vector<ElemBasis> v = compute_v();
          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma);
          Vector<ElemBasis> J = compute_AxB(normal,B);
          Vector<ElemBasis> A;
          S.quadrature_FxU.Eval(A, S.GetElemList(), J, S.Laplace_FxU);
          Real flux = compute_inner_prod(v, A);

          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              S.Elem(i,0)[j] = X_orig[i*COORD_DIM+0][j];
              S.Elem(i,1)[j] = X_orig[i*COORD_DIM+1][j];
              S.Elem(i,2)[j] = X_orig[i*COORD_DIM+2][j];
            }
          }
          compute_norm_area_elem(normal, area_elem);
          S.quadrature_FxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxU , order_singular, order_direct, -1.0, comm);
          S.quadrature_FxdU.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxdU, order_singular, order_direct, -1.0, comm);

          return flux;
        };
        Real dphi_dnu = (compute_flux(nu,1e-3)-compute_flux(nu,-1e-3)) / 2e-3;
        std::cout<<"dphi_dnu = "<<dphi_dnu<<'\n';

        Real phi = compute_flux(nu,0);
        std::cout<<"phi = "<<phi<<'\n';
      }

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      auto compute_Av = [&S,&area_elem,&normal,&compute_norm_area_elem,&compute_A,&comm] (const Vector<Real>& v, const Vector<ElemBasis>& nu, Real eps) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> X_orig(Nelem*COORD_DIM);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            X_orig[i*COORD_DIM+0][j] = S.Elem(i,0)[j];
            X_orig[i*COORD_DIM+1][j] = S.Elem(i,1)[j];
            X_orig[i*COORD_DIM+2][j] = S.Elem(i,2)[j];
            S.Elem(i,0)[j] += eps*nu[i][j] * normal[i*COORD_DIM+0][j];
            S.Elem(i,1)[j] += eps*nu[i][j] * normal[i*COORD_DIM+1][j];
            S.Elem(i,2)[j] += eps*nu[i][j] * normal[i*COORD_DIM+2][j];
          }
        }
        compute_norm_area_elem(normal, area_elem);
        S.quadrature_FxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxU , order_singular, order_direct, -1.0, comm);
        S.quadrature_DxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_DxU , order_singular, order_direct, -1.0, comm);
        S.quadrature_FxdU.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxdU, order_singular, order_direct, -1.0, comm);
        S.quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm);

        Vector<Real> Av = compute_A(v);

        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            S.Elem(i,0)[j] = X_orig[i*COORD_DIM+0][j];
            S.Elem(i,1)[j] = X_orig[i*COORD_DIM+1][j];
            S.Elem(i,2)[j] = X_orig[i*COORD_DIM+2][j];
          }
        }
        compute_norm_area_elem(normal, area_elem);
        S.quadrature_FxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxU , order_singular, order_direct, -1.0, comm);
        S.quadrature_DxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_DxU , order_singular, order_direct, -1.0, comm);
        S.quadrature_FxdU.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxdU, order_singular, order_direct, -1.0, comm);
        S.quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm);

        return Av;
      };
      auto compute_u_dAdnu_v = [&S,&compute_Av,&compute_inner_prod] (const Vector<Real>& u, const Vector<Real>& v, const Vector<ElemBasis>& nu) {
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Real eps = 1e-5;
        Vector<Real> Av0 = compute_Av(v,nu,-eps);
        Vector<Real> Av1 = compute_Av(v,nu,eps);
        Vector<Real> dAdnu_v = (Av1-Av0)*(1/(2*eps));

        Real u_dAdnu_v;
        { // set u_dAdnu_v
          Vector<ElemBasis> u_(Nelem), dAdnu_v_(Nelem);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              u_[i][j] = u[i*Nnodes+j];
              dAdnu_v_[i][j] = dAdnu_v[i*Nnodes+j];
            }
          }
          u_dAdnu_v = compute_inner_prod(u_, dAdnu_v_);
          u_dAdnu_v += u[Nelem*Nnodes] * dAdnu_v[Nelem*Nnodes];
        }
        return u_dAdnu_v;
      };
      if (0) { // test dA_dnu
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> nu(Nelem);
        Vector<Real> u(Nelem*Nnodes+1), v(Nelem*Nnodes+1);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            v[i*Nnodes+j] = sigma[i][j];
            u[i*Nnodes+j] = sigma[i][j]*area_elem[i][j];
          }
        }
        v[Nelem*Nnodes] = 0; //alpha;
        u[Nelem*Nnodes] = 0;
        nu = 1; //area_elem;

        Real u_dAdnu_v = compute_u_dAdnu_v(u, v, nu);
        std::cout<<"u_dAdnu_v = "<<u_dAdnu_v<<'\n';
      }

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      auto compute_dsigma_dnu = [&S,&area_elem,&normal,&compute_norm_area_elem,&compute_invA,&comm] (const Vector<ElemBasis>& nu, Real eps) {
        auto compute_sigma = [&S,&area_elem,&normal,&compute_norm_area_elem,&compute_invA,&comm] (const Vector<ElemBasis>& nu, Real eps) {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> X_orig(Nelem*COORD_DIM);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              X_orig[i*COORD_DIM+0][j] = S.Elem(i,0)[j];
              X_orig[i*COORD_DIM+1][j] = S.Elem(i,1)[j];
              X_orig[i*COORD_DIM+2][j] = S.Elem(i,2)[j];
              S.Elem(i,0)[j] += eps*nu[i][j] * normal[i*COORD_DIM+0][j];
              S.Elem(i,1)[j] += eps*nu[i][j] * normal[i*COORD_DIM+1][j];
              S.Elem(i,2)[j] += eps*nu[i][j] * normal[i*COORD_DIM+2][j];
            }
          }
          compute_norm_area_elem(normal, area_elem);
          S.quadrature_FxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxU , order_singular, order_direct, -1.0, comm);
          S.quadrature_DxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_DxU , order_singular, order_direct, -1.0, comm);
          S.quadrature_FxdU.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxdU, order_singular, order_direct, -1.0, comm);
          S.quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm);

          Real flux = 1.0, alpha;
          Vector<ElemBasis> sigma;
          compute_invA(sigma, alpha, flux);
          Vector<Real> sigma_(Nelem*Nnodes+1);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              sigma_[i*Nnodes+j] = sigma[i][j];
            }
          }
          sigma_[Nelem*Nnodes] = alpha;

          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              S.Elem(i,0)[j] = X_orig[i*COORD_DIM+0][j];
              S.Elem(i,1)[j] = X_orig[i*COORD_DIM+1][j];
              S.Elem(i,2)[j] = X_orig[i*COORD_DIM+2][j];
            }
          }
          compute_norm_area_elem(normal, area_elem);
          S.quadrature_FxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxU , order_singular, order_direct, -1.0, comm);
          S.quadrature_DxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_DxU , order_singular, order_direct, -1.0, comm);
          S.quadrature_FxdU.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxdU, order_singular, order_direct, -1.0, comm);
          S.quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm);

          return sigma_;
        };
        auto sigma0 = compute_sigma(nu,-eps);
        auto sigma1 = compute_sigma(nu,eps);
        return (sigma1-sigma0) * (1/(2*eps));
      };
      if (0) { // verify dA_dnu sigma + A dsigma_dnu = 0
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> nu(Nelem);
        nu = 1; //area_elem;

        Vector<Real> dA_dnu_sigma;
        { // Set dA_dnu_simga
          Vector<Real> sigma_(Nelem*Nnodes+1);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              sigma_[i*Nnodes+j] = sigma[i][j];
            }
          }
          sigma_[Nelem*Nnodes] = alpha;

          Real eps = 1e-3;
          Vector<Real> Asigma0 = compute_Av(sigma_,nu,-eps);
          Vector<Real> Asigma1 = compute_Av(sigma_,nu,eps);
          dA_dnu_sigma = (Asigma1-Asigma0) * (1/(2*eps));
        }

        Vector<Real> A_dsigma_dnu;
        { // Set A_dsigma_dnu
          Vector<Real> dsigma_dnu = compute_dsigma_dnu(nu, 1e-3);
          A_dsigma_dnu = compute_A(dsigma_dnu);
        }

        Vector<ElemBasis> dA_dnu_sigma_(Nelem);
        Vector<ElemBasis> A_dsigma_dnu_(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            dA_dnu_sigma_[i][j] = dA_dnu_sigma[i*Nnodes+j];
            A_dsigma_dnu_[i][j] = A_dsigma_dnu[i*Nnodes+j];
          }
        }
        std::cout<<dA_dnu_sigma[Nelem*Nnodes] + A_dsigma_dnu[Nelem*Nnodes]<<'\n';

        { // Write VTU
          VTUData vtu;
          vtu.AddElems(S.GetElemList(), dA_dnu_sigma_ + A_dsigma_dnu_, ORDER);
          vtu.WriteVTK("err", comm);
        }
      }

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      if (1) { // test grad_g
        const Long Nelem = S.GetElemList().NElem();
        const Long Nnodes = ElemBasis::Size();

        Vector<ElemBasis> dg_dnu;
        { // Compute dg_dnu
          dg_dnu = compute_dg_dnu(sigma, alpha, B);

          Vector<Real> dg_dsigma(Nelem*Nnodes+1);
          { // Set dg_dsigma
            Vector<ElemBasis> dg_dsigma_ = compute_dg_dsigma(B);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                dg_dsigma[i*Nnodes+j] = dg_dsigma_[i][j];
              }
            }
            dg_dsigma[Nelem*Nnodes] = compute_dg_dalpha(B);
          }

          Vector<Real> dg_dsigma_invA = compute_invAadj(dg_dsigma);

          { // Set dg_dnu = - dg_dsigma invA dA_dnu sigma
            Vector<Real> sigma_(Nelem*Nnodes+1);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                sigma_[i*Nnodes+j] = sigma[i][j];
              }
            }
            sigma_[Nelem*Nnodes] = alpha;

            auto dg_dnu1 = compute_u_dAdnu_v_00(dg_dsigma_invA, sigma_)*(-1);
            auto dg_dnu2 = compute_u_dAdnu_v_01(dg_dsigma_invA, sigma_)*(-1);
            auto dg_dnu3 = compute_u_dAdnu_v_10(dg_dsigma_invA, sigma_)*(-1);
            auto dg_dnu4 = compute_u_dAdnu_v_11(dg_dsigma_invA, sigma_)*(-1);

            {
              //Vector<ElemBasis> nu(Nelem);
              //nu = area_elem;

              //Real dg_dnu0_ = -compute_inner_prod(nu, dg_dnu);
              //Real dg_dnu1_ = -compute_inner_prod(nu, dg_dnu1);
              //Real dg_dnu2_ = -compute_inner_prod(nu, dg_dnu2);
              //Real dg_dnu3_ = -compute_inner_prod(nu, dg_dnu3);
              //Real dg_dnu4_ = -compute_inner_prod(nu, dg_dnu4);
              //std::cout<<dg_dnu0_<<' '<<dg_dnu1_<<' '<<dg_dnu2_<<' '<<dg_dnu3_<<' '<<dg_dnu4_<<'\n';
            }

            dg_dnu += dg_dnu1;
            dg_dnu += dg_dnu2;
            dg_dnu += dg_dnu3;
            dg_dnu += dg_dnu4;
          }

          if (0) { // Set dg_dnu = - dg_dsigma invA dA_dnu sigma
            Vector<ElemBasis> nu(Nelem);
            nu = dg_dnu; //1; //area_elem;

            Vector<Real> dg_dsigma_invA = compute_invAadj(dg_dsigma);
            Vector<Real> sigma_(Nelem*Nnodes+1);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                sigma_[i*Nnodes+j] = sigma[i][j];
              }
            }
            sigma_[Nelem*Nnodes] = alpha;

            Vector<Real> dg_dsigma_invA_0 = dg_dsigma_invA; dg_dsigma_invA_0[Nelem*Nnodes] = 0;
            Vector<Real> dg_dsigma_invA_1(Nelem*Nnodes+1); dg_dsigma_invA_1 = 0; dg_dsigma_invA_1[Nelem*Nnodes] = dg_dsigma_invA[Nelem*Nnodes];

            Vector<Real> sigma_0 = sigma_; sigma_0[Nelem*Nnodes] = 0;
            Vector<Real> sigma_1(Nelem*Nnodes+1); sigma_1 = 0; sigma_1[Nelem*Nnodes] = sigma_[Nelem*Nnodes];

            Real dg_dnu1 = -compute_u_dAdnu_v(dg_dsigma_invA_0, sigma_0, nu);
            Real dg_dnu2 = -compute_u_dAdnu_v(dg_dsigma_invA_0, sigma_1, nu);
            Real dg_dnu3 = -compute_u_dAdnu_v(dg_dsigma_invA_1, sigma_0, nu);
            Real dg_dnu4 = -compute_u_dAdnu_v(dg_dsigma_invA_1, sigma_1, nu);
            std::cout<<dg_dnu1<<' '<<dg_dnu2<<' '<<dg_dnu3<<' '<<dg_dnu4<<'\n';
          }
          if (0) { // Set dg_dnu = dg_dsigma dsigma_dnu
            Vector<ElemBasis> nu(Nelem);
            nu = dg_dnu; //1; //area_elem;

            Vector<Real> dsigma_dnu = compute_dsigma_dnu(nu, 1e-3);
            Vector<ElemBasis> dg_dsigma_(Nelem), dsigma_dnu_(Nelem);
            for (Long i = 0; i < Nelem; i++) {
              for (Long j = 0; j < Nnodes; j++) {
                dg_dsigma_[i][j] = dg_dsigma[i*Nnodes+j];
                dsigma_dnu_[i][j] = dsigma_dnu[i*Nnodes+j];
              }
            }
            Real dg_dnu = compute_inner_prod(dg_dsigma_, dsigma_dnu_);
            dg_dnu += dg_dsigma[Nelem*Nnodes] * dsigma_dnu[Nelem*Nnodes];
            std::cout<<dg_dnu<<'\n';
          }
        }
        { // Write VTU
          VTUData vtu;
          vtu.AddElems(S.GetElemList(), dg_dnu, ORDER);
          vtu.WriteVTK("dg_dnu", comm);
        }
        { // filter dg_dnu and write VTU
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();
          const Integer INTERP_ORDER = 12;
          Long Nt = S.NtNp_[0]*ORDER/3, Np = S.NtNp_[1]*ORDER/3;

          Matrix<Real> M(Nt, Np); M = 0;
          const auto& quad_wts = ElemBasis::QuadWts();
          const Matrix<Real>& Mnodes = Basis<Real,1,ORDER>::Nodes();
          for (Long tt = 0; tt < S.NtNp_[0]; tt++) {
            for (Long pp = 0; pp < S.NtNp_[1]; pp++) {
              for (Long t = 0; t < ORDER; t++) {
                for (Long p = 0; p < ORDER; p++) {
                  Real theta = (tt + Mnodes[0][t]) / S.NtNp_[0];
                  Real phi   = (pp + Mnodes[0][p]) / S.NtNp_[1];
                  Long i = (Long)(theta * Nt);
                  Long j = (Long)(phi   * Np);
                  Real x = theta * Nt - i;
                  Real y = phi   * Np - j;

                  Long elem_idx = tt * S.NtNp_[1] + pp;
                  Long node_idx = p * ORDER + t;

                  Vector<Real> Interp0(INTERP_ORDER);
                  Vector<Real> Interp1(INTERP_ORDER);
                  { // Set Interp0, Interp1
                    auto node = [] (Long i) {
                      return (Real)i - (INTERP_ORDER-1)/2;
                    };
                    for (Long i = 0; i < INTERP_ORDER; i++) {
                      Real wt_x = 1, wt_y = 1;
                      for (Long j = 0; j < INTERP_ORDER; j++) {
                        if (j != i) {
                          wt_x *= (x - node(j)) / (node(i) - node(j));
                          wt_y *= (y - node(j)) / (node(i) - node(j));
                        }
                        Interp0[i] = wt_x;
                        Interp1[i] = wt_y;
                      }
                    }
                  }

                  for (Long ii = 0; ii < INTERP_ORDER; ii++) {
                    for (Long jj = 0; jj < INTERP_ORDER; jj++) {
                      Long idx_i = (i + ii-(INTERP_ORDER-1)/2 + Nt) % Nt;
                      Long idx_j = (j + jj-(INTERP_ORDER-1)/2 + Np) % Np;
                      M[idx_i][idx_j] += dg_dnu[elem_idx][node_idx] * quad_wts[node_idx] * Interp0[ii] * Interp1[jj] / (S.NtNp_[0] * S.NtNp_[1]) * (Nt * Np);
                    }
                  }
                }
              }
            }
          }

          Vector<ElemBasis> f(Nelem);
          for (Long tt = 0; tt < S.NtNp_[0]; tt++) {
            for (Long pp = 0; pp < S.NtNp_[1]; pp++) {
              for (Long t = 0; t < ORDER; t++) {
                for (Long p = 0; p < ORDER; p++) {
                  Matrix<Real> Mnodes = Basis<Real,1,ORDER>::Nodes();
                  Real theta = (tt + Mnodes[0][t]) / S.NtNp_[0];
                  Real phi   = (pp + Mnodes[0][p]) / S.NtNp_[1];
                  Long i = (Long)(theta * Nt);
                  Long j = (Long)(phi   * Np);
                  Real x = theta * Nt - i;
                  Real y = phi   * Np - j;

                  Vector<Real> Interp0(INTERP_ORDER);
                  Vector<Real> Interp1(INTERP_ORDER);
                  { // Set Interp0, Interp1
                    auto node = [] (Long i) {
                      return (Real)i - (INTERP_ORDER-1)/2;
                    };
                    for (Long i = 0; i < INTERP_ORDER; i++) {
                      Real wt_x = 1, wt_y = 1;
                      for (Long j = 0; j < INTERP_ORDER; j++) {
                        if (j != i) {
                          wt_x *= (x - node(j)) / (node(i) - node(j));
                          wt_y *= (y - node(j)) / (node(i) - node(j));
                        }
                        Interp0[i] = wt_x;
                        Interp1[i] = wt_y;
                      }
                    }
                  }

                  Real f0 = 0;
                  for (Long ii = 0; ii < INTERP_ORDER; ii++) {
                    for (Long jj = 0; jj < INTERP_ORDER; jj++) {
                      Long idx_i = (i + ii-(INTERP_ORDER-1)/2 + Nt) % Nt;
                      Long idx_j = (j + jj-(INTERP_ORDER-1)/2 + Np) % Np;
                      f0 += Interp0[ii] * Interp1[jj] * M[idx_i][idx_j];
                    }
                  }

                  Long elem_idx = tt * S.NtNp_[1] + pp;
                  Long node_idx = p * ORDER + t;
                  f[elem_idx][node_idx] = f0;
                }
              }
            }
          }
          { // Write VTU
            VTUData vtu;
            vtu.AddElems(S.GetElemList(), f, ORDER);
            vtu.WriteVTK("dg_dnu_filtered", comm);
          }
        }

        auto compute_g = [&sigma,&alpha,&S,&area_elem,&normal,&compute_norm_area_elem,&compute_invA,&compute_half_n_plus_dG,&compute_B0,&compute_inner_prod,&comm] (const Vector<ElemBasis>& nu, Real eps) {
          const Long Nelem = S.GetElemList().NElem();
          const Long Nnodes = ElemBasis::Size();

          Vector<ElemBasis> X_orig(Nelem*COORD_DIM);
          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              X_orig[i*COORD_DIM+0][j] = S.Elem(i,0)[j];
              X_orig[i*COORD_DIM+1][j] = S.Elem(i,1)[j];
              X_orig[i*COORD_DIM+2][j] = S.Elem(i,2)[j];
              S.Elem(i,0)[j] += eps*nu[i][j] * normal[i*COORD_DIM+0][j];
              S.Elem(i,1)[j] += eps*nu[i][j] * normal[i*COORD_DIM+1][j];
              S.Elem(i,2)[j] += eps*nu[i][j] * normal[i*COORD_DIM+2][j];
            }
          }
          compute_norm_area_elem(normal, area_elem);
          S.quadrature_FxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxU , order_singular, order_direct, -1.0, comm);
          S.quadrature_DxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_DxU , order_singular, order_direct, -1.0, comm);
          S.quadrature_FxdU.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxdU, order_singular, order_direct, -1.0, comm);
          S.quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm);

          Real flux = 1.0, alpha;
          Vector<ElemBasis> sigma;
          compute_invA(sigma, alpha, flux);
          Vector<ElemBasis> B = compute_half_n_plus_dG(sigma) + compute_B0(alpha);
          Real g = compute_inner_prod(B, B);

          for (Long i = 0; i < Nelem; i++) {
            for (Long j = 0; j < Nnodes; j++) {
              S.Elem(i,0)[j] = X_orig[i*COORD_DIM+0][j];
              S.Elem(i,1)[j] = X_orig[i*COORD_DIM+1][j];
              S.Elem(i,2)[j] = X_orig[i*COORD_DIM+2][j];
            }
          }
          compute_norm_area_elem(normal, area_elem);
          S.quadrature_FxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxU , order_singular, order_direct, -1.0, comm);
          S.quadrature_DxU .template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_DxU , order_singular, order_direct, -1.0, comm);
          S.quadrature_FxdU.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_FxdU, order_singular, order_direct, -1.0, comm);
          S.quadrature_dUxF.template Setup<ElemBasis, ElemBasis>(S.GetElemList(), S.Laplace_dUxF, order_singular, order_direct, -1.0, comm);

          return g;
        };

        {
          Vector<ElemBasis> nu(Nelem);
          nu = area_elem;

          Real eps = 1e-4;
          Real g0 = compute_g(nu,-eps);
          Real g1 = compute_g(nu,eps);
          std::cout<<"g = "<<g0<<"  g = "<<g1<<"  dg_dnu = "<<(g1-g0)/(2*eps)<<'\n';
          std::cout<<"dg_dnu = "<<compute_inner_prod(nu, dg_dnu)<<'\n';
        }
        {
          Vector<ElemBasis> nu(Nelem);
          nu = 1;

          Real eps = 1e-4;
          Real g0 = compute_g(nu,-eps);
          Real g1 = compute_g(nu,eps);
          std::cout<<"g = "<<g0<<"  g = "<<g1<<"  dg_dnu = "<<(g1-g0)/(2*eps)<<'\n';
          std::cout<<"dg_dnu = "<<compute_inner_prod(nu, dg_dnu)<<'\n';
        }
        {
          Vector<ElemBasis> nu(Nelem);
          nu = dg_dnu;

          Real eps = 1e-4;
          Real g0 = compute_g(nu,-eps);
          Real g1 = compute_g(nu,eps);
          std::cout<<"g = "<<g0<<"  g = "<<g1<<"  dg_dnu = "<<(g1-g0)/(2*eps)<<'\n';
          std::cout<<"dg_dnu = "<<compute_inner_prod(nu, dg_dnu)<<'\n';
        }
      }

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      // dg_dnu

      // dA_dnu_sigma

      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
      /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////

      //Profile::print(&comm);
    }

  private:
    void InitSurf(Long l) {
      const auto& nodes = ElemBasis::Nodes();
      const Long Nt = NtNp_[l*2+0];
      const Long Np = NtNp_[l*2+1];

      for (Long i = 0; i < Nt; i++) {
        for (Long j = 0; j < Np; j++) {
          for (Long k = 0; k < ElemBasis::Size(); k++) {
            Real theta = (i + nodes[0][k]) * 2*const_pi<Real>()/Nt;
            Real phi   = (j + nodes[1][k]) * 2*const_pi<Real>()/Np;
            Real X,Y,Z;
            SurfGeom(X,Y,Z,theta,phi);
            Elem(ElemIdx(l,i,j),0)[k] = X;
            Elem(ElemIdx(l,i,j),1)[k] = Y;
            Elem(ElemIdx(l,i,j),2)[k] = Z;
          }
        }
      }
    }

    static void SurfGeom(Real& X, Real& Y, Real& Z, Real theta, Real phi) {
      sctl::Integer Nperiod = 5;
#if 0
      Real Aspect_ratio = 10.27932548522949;
      Real coeffmat[21][21] = { 0.00000478813217,  0.00000000000000,  0.00000351611652,  0.00000135354389,  0.00000061357832,  0.00000220091101,  0.00000423862912, -0.00003000058678,  0.00000064187111, -0.00024228452821,  0.00003116775770,  0.00000176210710,  0.00000289141326, -0.00000150300525,  0.00000772853855,  0.00000098855242,  0.00000316606793,  0.00000002168364,  0.00000212047939,  0.00000299016097,  0.00000443224508,
                                0.00000028202930,  0.00000000000000, -0.00000249222421, -0.00000203136278,  0.00000131104809,  0.00000011987446, -0.00000370760154,  0.00004553918916, -0.00007711342914, -0.00004685295062,  0.00011049838213, -0.00000197486270,  0.00000395827146,  0.00000615046474,  0.00000755337123,  0.00000700606006,  0.00000922725030, -0.00000043310337,  0.00000107416383,  0.00000449787694,  0.00000305137178,
                                0.00001226376662,  0.00000000000000,  0.00000270820692,  0.00000208059305,  0.00000521478523,  0.00001779037302,  0.00000846544117,  0.00001120913385, -0.00065816845745, -0.00085107452469, -0.00013171190221, -0.00005540943675, -0.00001835885450,  0.00000101879823,  0.00000209222071,  0.00000091532502, -0.00000521515358, -0.00000209227142, -0.00000678545939, -0.00000034963549, -0.00000015111488,
                                0.00001560274177,  0.00000000000000,  0.00000350691471, -0.00001160475040, -0.00001763036562,  0.00003487367940, -0.00002787247831, -0.00000910982726,  0.00008818832430, -0.00524408789352,  0.00009378376126,  0.00004184526188,  0.00002849263365, -0.00002757280527,  0.00003388467667,  0.00000706207265,  0.00000625263419, -0.00003315929280, -0.00001181772132,  0.00000311426015,  0.00001875682574,
                               -0.00000398287420,  0.00000000000000, -0.00001524541040,  0.00001724056165,  0.00002245173346,  0.00002806861812, -0.00000388776925,  0.00008143573359, -0.00005900909309,  0.00110496615525,  0.00134626252111,  0.00005128383054, -0.00001372421866,  0.00003612563887,  0.00002236580076, -0.00002728391883,  0.00001981237256,  0.00000655450458,  0.00000985319002,  0.00001347597299,  0.00000645987802,
                                0.00003304968050,  0.00000000000000, -0.00000530822217,  0.00001324870937, -0.00003610889689, -0.00005478735329, -0.00005818806312, -0.00037112057908, -0.00017812002625, -0.00093204283621,  0.00115969858598, -0.00033559172880, -0.00010441876657, -0.00001617923044, -0.00000555065844,  0.00007343527250, -0.00004408047607,  0.00000403802142,  0.00001843931204,  0.00001694047933,  0.00001213414362,
                               -0.00000751115658,  0.00000000000000,  0.00005457974839, -0.00000334614515,  0.00005845565465,  0.00015000770509,  0.00021849104087,  0.00002724147635,  0.00167233624961,  0.00011666602222,  0.00276563479565, -0.00085952825611, -0.00030217235326, -0.00008841593808,  0.00000997664119, -0.00015285826521,  0.00002517224675,  0.00003009161810,  0.00001883217556,  0.00002146127554,  0.00001822445302,
                               -0.00004128706860,  0.00000000000000, -0.00003496417776,  0.00001088761655, -0.00000298955979, -0.00005359326315, -0.00019021633489, -0.00017992728681, -0.00347794801928,  0.00064632791327,  0.00449698418379, -0.00017710507382,  0.00006126180233,  0.00018059254216,  0.00002354096432,  0.00008189838991, -0.00010060678323, -0.00017183290038,  0.00019413756672,  0.00021334811754,  0.00011263617489,
                                0.00000853522670, -0.00000000000000, -0.00006544789358,  0.00005424076880, -0.00000679056529, -0.00001249735487, -0.00053082982777,  0.00035396864405, -0.00115020677913,  0.05894451215863,  0.06573092192411,  0.01498018857092,  0.00278125284240,  0.00145188067108,  0.00033717858605,  0.00000800427370, -0.00009335305367,  0.00024286781263, -0.00023916347709,  0.00031213948387,  0.00018134393031,
                               -0.00002521496390, -0.00000000000000, -0.00054337945767,  0.00012690725271,  0.00053313979879,  0.00064233405283, -0.00047686311882,  0.00176536326762,  0.00074157933705, -0.02684566564858,  1.00000000000000,  0.07176169008017,  0.00837037432939, -0.00000381640211,  0.00088998704450, -0.00049218931235, -0.00024546548957, -0.00036608282244,  0.00049480766756,  0.00031158892671,  0.00006898906577,
                                0.00021280418150,  0.00028127161204, -0.00070030166535,  0.00022237010126, -0.00028713891516, -0.00013800295710,  0.00005912094275,  0.00172126013786, -0.00618684850633,  0.03608432412148,    Aspect_ratio  ,  0.49896776676178,  0.00091372377938, -0.00085712829605, -0.00124801427592, -0.00007427225501, -0.00005245858847,  0.00002841771493,  0.00020249813679, -0.00014303345233,  0.00001406490901,
                                0.00023699452868,  0.00008661757602,  0.00025744654704, -0.00022715188970, -0.00076146807987,  0.00055185536621, -0.00012325309217, -0.00072356045712, -0.00160693109501,  0.00246682553552, -0.14175094664097, -0.36207047104836, -0.04089594259858,  0.00060774467420,  0.00088646943914,  0.00004865296432, -0.00041878610500, -0.00023025234987, -0.00009676301852, -0.00000000000000,  0.00008409228758,
                                0.00011432896281, -0.00000707848403,  0.00004698805787, -0.00043642931269,  0.00081384339137, -0.00065635429928, -0.00011831733718,  0.00017413357273,  0.00224463525228,  0.00478497287259,  0.03294761106372,  0.01078986655921,  0.10731782764196,  0.00075034319889, -0.00009241879889,  0.00055023463210,  0.00006596000458,  0.00005045382932,  0.00014874986664,  0.00000000000000, -0.00015369028552,
                                0.00001037383754,  0.00009250180301,  0.00026204055757,  0.00007424291834, -0.00047751804232,  0.00029184055165,  0.00050921301590, -0.00004825839278, -0.00029933769838,  0.00279659987427,  0.00210463814437, -0.00618590926751, -0.02400829829276, -0.02316811867058, -0.00086368201301, -0.00032258985448, -0.00018304496189,  0.00008438774967, -0.00008305341908,  0.00000000000000,  0.00013047417451,
                               -0.00001376930322, -0.00001723831701, -0.00011543079017, -0.00022646733851,  0.00013467084500, -0.00004661652201, -0.00008419520600,  0.00035772417323, -0.00011815709877,  0.00028718306567,  0.00092207465786, -0.00317224999890,  0.00061770365573,  0.01017294172198,  0.00294739892706,  0.00014669894881,  0.00015702951350,  0.00003432080121, -0.00008555022214, -0.00000000000000,  0.00000454909878,
                               -0.00000196001542, -0.00003198397462, -0.00004425687075, -0.00004129848094, -0.00003789070615, -0.00027583551127,  0.00025874207495, -0.00002334945384, -0.00007259396807, -0.00008295358566,  0.00011360697681, -0.00101968157105,  0.00046784928418, -0.00208410434425, -0.00313158822246, -0.00046005158219, -0.00010552268213, -0.00005850767775,  0.00003971093611,  0.00000000000000, -0.00005275657168,
                               -0.00001065901233, -0.00001934838656, -0.00001220186732, -0.00002060524639, -0.00000225423423, -0.00001894621164, -0.00001533334580, -0.00001791087379,  0.00008156246622, -0.00008441298269,  0.00021060956351, -0.00030303673702,  0.00075949780876, -0.00010539998038,  0.00109045265708,  0.00068949378328,  0.00009268362192,  0.00003471063246,  0.00001204656473, -0.00000000000000,  0.00001500743110,
                                0.00000105878155, -0.00000910870767, -0.00000172467264, -0.00000722095228,  0.00000699280463, -0.00002061720625, -0.00000889817693, -0.00001993474507,  0.00000370749740, -0.00000090311920,  0.00002677819793,  0.00043428712524,  0.00210293265991,  0.00018200518389, -0.00009621794743, -0.00035250501242, -0.00012996385340, -0.00002185157609, -0.00001116586463, -0.00000000000000, -0.00000451994811,
                                0.00000424055270, -0.00000463139304,  0.00000301006116, -0.00000123974939,  0.00000632465435, -0.00002090823000,  0.00001773388794,  0.00000121050368,  0.00001886057362, -0.00001043497195, -0.00002269273500, -0.00021979617304, -0.00001043962493, -0.00116343051195, -0.00004193381756,  0.00007944958634,  0.00007301353617,  0.00002082651736, -0.00000119863023, -0.00000000000000, -0.00001440504820,
                               -0.00000391270805, -0.00000490489265, -0.00000504441778, -0.00000904507579, -0.00000111389932,  0.00000597532107,  0.00000047090245, -0.00001553130096, -0.00001524566323, -0.00000522222899, -0.00007707672921, -0.00004165665086,  0.00015764687851,  0.00035649110214,  0.00038701237645,  0.00002386798405, -0.00001946414341, -0.00000913835174, -0.00000489907188,  0.00000000000000,  0.00000172327657,
                               -0.00000015388650, -0.00000603232729, -0.00000397650865,  0.00000280493782,  0.00000463132073, -0.00000788678426, -0.00000471605335, -0.00000283715985, -0.00000422824724,  0.00000366817630, -0.00001159603562, -0.00001625759251,  0.00049116823357,  0.00005048640014, -0.00020234247495, -0.00006341376866, -0.00000807822744,  0.00000070463199,  0.00000014041755,  0.00000000000000, -0.00000718306910};
#else
      Real Aspect_ratio = 5;
      Real coeffmat[21][21] = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            1, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Aspect_ratio, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0.2, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0,
                               0, 0, 0, 0, 0, 0, 0, 0, 0, 0,            0, 0  , 0, 0, 0, 0, 0, 0, 0, 0, 0};
#endif
      Z = 0;
      Real R = 0;
      for (long i = -10; i <= 10; i++) {
        for (long j = -10; j <= 10; j++) {
          R += coeffmat[i+10][j+10] * sctl::cos(-i*phi + Nperiod*j*theta);
          Z += coeffmat[i+10][j+10] * sctl::sin(-i*phi + Nperiod*j*theta);
        }
      }
      X = R * sctl::cos(theta);
      Y = R * sctl::sin(theta);
    }

    GenericKernel<BiotSavart3D  > BiotSavart  ;
    GenericKernel<Laplace3D_FxU > Laplace_FxU ;
    GenericKernel<Laplace3D_DxU > Laplace_DxU ;
    GenericKernel<Laplace3D_FxdU> Laplace_FxdU;
    GenericKernel<Laplace3D_dUxF> Laplace_dUxF;
    GenericKernel<Laplace3D_Fxd2U> Laplace_Fxd2U;
    GenericKernel<Laplace3D_dUxD> Laplace_dUxD;
    GenericKernel<Laplace3D_DxdU> Laplace_DxdU;
    Quadrature<Real> quadrature_FxU ;
    Quadrature<Real> quadrature_DxU ;
    Quadrature<Real> quadrature_FxdU;
    Quadrature<Real> quadrature_dUxF;
    Quadrature<Real> quadrature_Fxd2U;
    Quadrature<Real> quadrature_dUxD;

    ElemLst elements;
    Vector<Long> NtNp_;
    Vector<Long> elem_dsp;
};

template <class Real, Integer ORDER=5> class Spheres {
  static constexpr Integer COORD_DIM = 3;
  static constexpr Integer ELEM_DIM = COORD_DIM-1;
  using PotentialBasis = Basis<Real, ELEM_DIM, ORDER>;
  using DensityBasis = Basis<Real, ELEM_DIM, ORDER>;
  using CoordBasis = Basis<Real, ELEM_DIM, ORDER>;
  using ElemLst = ElemList<COORD_DIM, CoordBasis>;

  public:
    Spheres(Long N = 0) {
      Vector<Real> X(N*COORD_DIM);
      Vector<Real> R(N);
      X=0;
      R=1;
      for (Long i = 0; i < N; i++) X[i*COORD_DIM] = (i==0?-1.015:1.015); ///////////
      InitSpheres(X,R);
    }

    const ElemLst& GetElem() const {
      return elements;
    }

    static void test() {
      constexpr Integer order_singular = 35;
      constexpr Integer order_direct = 35;
      Comm comm = Comm::World();
      Profile::Enable(true);

      Long Ns = 2;
      Spheres S(Ns);
      S.quadrature_FxT.template Setup<DensityBasis, PotentialBasis>(S.GetElem(), S.Stokes_FxT, order_singular, order_direct, -1.0, comm);
      S.quadrature_FxU.template Setup<DensityBasis, PotentialBasis>(S.GetElem(), S.Stokes_FxU, order_singular, order_direct, -1.0, comm);
      S.quadrature_DxU.template Setup<DensityBasis, PotentialBasis>(S.GetElem(), S.Stokes_DxU, order_singular, order_direct, -1.0, comm);

      const auto SetMotion = [&S](Vector<DensityBasis>& density, const Vector<Real>& force_avg, const Vector<Real>& torque_avg) {
        Long Nelem = S.GetElem().NElem();
        Long Nsurf = S.elem_cnt.Dim();
        const auto& X = S.GetElem().ElemVector();

        Vector<Real> area, Xc;
        Vector<DensityBasis> one(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < DensityBasis::Size(); j++) {
            one[i][j] = 1;
          }
        }
        S.SurfInteg(area, one);
        S.SurfInteg(Xc, S.GetElem().ElemVector());
        for (Long i = 0; i < Nsurf; i++) {
          for (Long k = 0; k < COORD_DIM; k++) {
            Xc[i*COORD_DIM+k] /= area[i];
          }
        }

        if (density.Dim() != Nelem*COORD_DIM) density.ReInit(Nelem*COORD_DIM);
        Long elem_itr = 0;
        for (Long i = 0; i < Nsurf; i++) {
          for (Long j = 0; j < S.elem_cnt[i]; j++) {
            for (Long k = 0; k < DensityBasis::Size(); k++) {
              StaticArray<Real,COORD_DIM> dX;
              dX[0] = (X[elem_itr*COORD_DIM+0][k] - Xc[i*COORD_DIM+0]);
              dX[1] = (X[elem_itr*COORD_DIM+1][k] - Xc[i*COORD_DIM+1]);
              dX[2] = (X[elem_itr*COORD_DIM+2][k] - Xc[i*COORD_DIM+2]);
              density[elem_itr*COORD_DIM+0][k] = force_avg[i*COORD_DIM+0]*(1/area[i]) + (torque_avg[i*COORD_DIM+1] * dX[2] - torque_avg[i*COORD_DIM+2] * dX[1]) / (2*area[i]/3);
              density[elem_itr*COORD_DIM+1][k] = force_avg[i*COORD_DIM+1]*(1/area[i]) + (torque_avg[i*COORD_DIM+2] * dX[0] - torque_avg[i*COORD_DIM+0] * dX[2]) / (2*area[i]/3);
              density[elem_itr*COORD_DIM+2][k] = force_avg[i*COORD_DIM+2]*(1/area[i]) + (torque_avg[i*COORD_DIM+0] * dX[1] - torque_avg[i*COORD_DIM+1] * dX[0]) / (2*area[i]/3);
            }
            elem_itr++;
          }
        }
      };
      const auto GetMotion = [&S](Vector<Real>& force_avg, Vector<Real>& torque_avg, const Vector<DensityBasis>& density) {
        Long Nelem = S.GetElem().NElem();
        Long Nsurf = S.elem_cnt.Dim();
        const auto& X = S.GetElem().ElemVector();

        S.SurfInteg(force_avg, density);

        Vector<Real> area, Xc;
        Vector<DensityBasis> one(Nelem);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < DensityBasis::Size(); j++) {
            one[i][j] = 1;
          }
        }
        S.SurfInteg(area, one);
        S.SurfInteg(Xc, S.GetElem().ElemVector());
        for (Long i = 0; i < Nsurf; i++) {
          for (Long k = 0; k < COORD_DIM; k++) {
            Xc[i*COORD_DIM+k] /= area[i];
          }
        }

        { // Set torque_avg
          Long elem_itr = 0;
          Vector<DensityBasis> torque(Nelem*COORD_DIM);
          for (Long i = 0; i < Nsurf; i++) {
            for (Long j = 0; j < S.elem_cnt[i]; j++) {
              for (Long k = 0; k < DensityBasis::Size(); k++) {
                StaticArray<Real,COORD_DIM> dX;
                dX[0] = (X[elem_itr*COORD_DIM+0][k] - Xc[i*COORD_DIM+0]);
                dX[1] = (X[elem_itr*COORD_DIM+1][k] - Xc[i*COORD_DIM+1]);
                dX[2] = (X[elem_itr*COORD_DIM+2][k] - Xc[i*COORD_DIM+2]);
                torque[elem_itr*COORD_DIM+0][k] = dX[1] * density[elem_itr*COORD_DIM+2][k] - dX[2] * density[elem_itr*COORD_DIM+1][k];
                torque[elem_itr*COORD_DIM+1][k] = dX[2] * density[elem_itr*COORD_DIM+0][k] - dX[0] * density[elem_itr*COORD_DIM+2][k];
                torque[elem_itr*COORD_DIM+2][k] = dX[0] * density[elem_itr*COORD_DIM+1][k] - dX[1] * density[elem_itr*COORD_DIM+0][k];
              }
              elem_itr++;
            }
          }
          S.SurfInteg(torque_avg, torque);
        }
      };
      const auto BIOpL = [&GetMotion,&SetMotion](Vector<DensityBasis>& potential, const Vector<DensityBasis>& density) {
        Vector<Real> force_avg, torque_avg;
        GetMotion(force_avg, torque_avg, density);
        SetMotion(potential, force_avg, torque_avg);
      };
      const auto BIOpK = [&S](Vector<DensityBasis>& potential, const Vector<DensityBasis>& density) {
        Vector<DensityBasis> traction;
        S.quadrature_FxT.Eval(traction, S.GetElem(), density, S.Stokes_FxT);

        Vector<CoordBasis> dX;
        const auto X = S.GetElem().ElemVector();
        CoordBasis::Grad(dX, X);

        Long Nelem = S.GetElem().NElem();
        Long Nnodes = CoordBasis::Size();
        potential.ReInit(Nelem * COORD_DIM);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            StaticArray<Real,COORD_DIM> Xn;
            Xn[0] = dX[i*COORD_DIM*2+2][j]*dX[i*COORD_DIM*2+5][j] - dX[i*COORD_DIM*2+4][j]*dX[i*COORD_DIM*2+3][j];
            Xn[1] = dX[i*COORD_DIM*2+4][j]*dX[i*COORD_DIM*2+1][j] - dX[i*COORD_DIM*2+0][j]*dX[i*COORD_DIM*2+5][j];
            Xn[2] = dX[i*COORD_DIM*2+0][j]*dX[i*COORD_DIM*2+3][j] - dX[i*COORD_DIM*2+2][j]*dX[i*COORD_DIM*2+1][j];
            Real AreaElem = sqrt<Real>(Xn[0]*Xn[0] + Xn[1]*Xn[1] + Xn[2]*Xn[2]);
            Real OOAreaElem = 1 / AreaElem;
            Xn[0] *= OOAreaElem;
            Xn[1] *= OOAreaElem;
            Xn[2] *= OOAreaElem;

            potential[i*COORD_DIM+0][j] = traction[i*COORD_DIM*COORD_DIM+0][j]*Xn[0] + traction[i*COORD_DIM*COORD_DIM+1][j]*Xn[1] + traction[i*COORD_DIM*COORD_DIM+2][j]*Xn[2];
            potential[i*COORD_DIM+1][j] = traction[i*COORD_DIM*COORD_DIM+3][j]*Xn[0] + traction[i*COORD_DIM*COORD_DIM+4][j]*Xn[1] + traction[i*COORD_DIM*COORD_DIM+5][j]*Xn[2];
            potential[i*COORD_DIM+2][j] = traction[i*COORD_DIM*COORD_DIM+6][j]*Xn[0] + traction[i*COORD_DIM*COORD_DIM+7][j]*Xn[1] + traction[i*COORD_DIM*COORD_DIM+8][j]*Xn[2];
          }
        }
      };
      const auto BIOp_half_K_L = [&S,&BIOpK,&BIOpL](Vector<DensityBasis>& potential, const Vector<DensityBasis>& density) {
        Vector<DensityBasis> potential_K;
        Vector<DensityBasis> potential_L;
        BIOpK(potential_K, density);
        BIOpL(potential_L, density);

        if (potential.Dim() != potential_K.Dim()) {
          potential.ReInit(potential_K.Dim());
        }
        for (Long i = 0; i < potential_K.Dim(); i++) {
          for (Long k = 0; k < DensityBasis::Size(); k++) {
            potential[i][k] = -0.5*density[i][k] + potential_K[i][k] + potential_L[i][k];
          }
        }
      };
      const auto BIOp_half_K = [&S,&BIOpK,&BIOpL](Vector<DensityBasis>& potential, const Vector<DensityBasis>& density) {
        Vector<DensityBasis> potential_K;
        BIOpK(potential_K, density);

        if (potential.Dim() != potential_K.Dim()) {
          potential.ReInit(potential_K.Dim());
        }
        for (Long i = 0; i < potential_K.Dim(); i++) {
          for (Long k = 0; k < DensityBasis::Size(); k++) {
            potential[i][k] = -0.5*density[i][k] + potential_K[i][k];
          }
        }
      };
      const auto BIOp_half_S_D = [&S,&BIOpL](Vector<DensityBasis>& potential, const Vector<DensityBasis>& density) {
        Vector<DensityBasis> U;
        S.quadrature_DxU.Eval(U, S.GetElem(), density, S.Stokes_DxU);

        Vector<PotentialBasis> U1;
        Vector<DensityBasis> sigma1;
        BIOpL(sigma1,density);
        S.quadrature_FxU.Eval(U1, S.GetElem(), sigma1, S.Stokes_FxU);

        Long Nelem = S.GetElem().NElem();
        Long Nnodes = CoordBasis::Size();
        potential.ReInit(Nelem * COORD_DIM);
        for (Long i = 0; i < Nelem; i++) {
          for (Long j = 0; j < Nnodes; j++) {
            potential[i*COORD_DIM+0][j] = 0.5*density[i*COORD_DIM+0][j] + U[i*COORD_DIM+0][j] + U1[i*COORD_DIM+0][j];
            potential[i*COORD_DIM+1][j] = 0.5*density[i*COORD_DIM+1][j] + U[i*COORD_DIM+1][j] + U1[i*COORD_DIM+1][j];
            potential[i*COORD_DIM+2][j] = 0.5*density[i*COORD_DIM+2][j] + U[i*COORD_DIM+2][j] + U1[i*COORD_DIM+2][j];
          }
        }
      };

      Vector<PotentialBasis> U;
      { // Rachh
        Vector<DensityBasis> sigma0;
        { // Set sigma0
          srand48(comm.Rank());
          Vector<Real> force(Ns*COORD_DIM), torque(Ns*COORD_DIM);
          //for (auto& x : force) x = drand48();
          //for (auto& x : torque) x = drand48();
          force = 0;
          torque = 0;
          force[0] = 1;
          //force[4] = 1;
          SetMotion(sigma0, force, torque);
        }

        Vector<DensityBasis> rhs;
        BIOp_half_K(rhs, sigma0);

        Vector<DensityBasis> sigma;
        { // Set sigma
          Long Nnode = DensityBasis::Size();
          Long Nelem = S.GetElem().NElem();

          typename sctl::ParallelSolver<Real>::ParallelOp A = [&S,&BIOp_half_K_L](sctl::Vector<Real>* Ax, const sctl::Vector<Real>& x) {
            Long Nnode = DensityBasis::Size();
            Long Nelem = S.GetElem().NElem();
            Ax->ReInit(Nelem*COORD_DIM*Nnode);

            Vector<DensityBasis> x_(Nelem*COORD_DIM), Ax_(Nelem*COORD_DIM);
            for (Long i = 0; i < Nelem*COORD_DIM; i++) { // Set x_
              for (Long k = 0; k < Nnode; k++) {
                x_[i][k] = x[i*Nnode+k];
              }
            }
            BIOp_half_K_L(Ax_, x_);
            for (Long i = 0; i < Nelem*COORD_DIM; i++) { // Set Ax
              for (Long k = 0; k < Nnode; k++) {
                (*Ax)[i*Nnode+k] = Ax_[i][k];
              }
            }
          };

          Vector<Real> sigma_(Nelem*COORD_DIM*Nnode), rhs_(Nelem*COORD_DIM*Nnode);
          for (Long i = 0; i < Nelem*COORD_DIM; i++) {// Set rhs_
            for (Long k = 0; k < Nnode; k++) {
              rhs_[i*Nnode+k] = rhs[i][k];
            }
          }
          sigma_ = 0;

          ParallelSolver<Real> linear_solver(comm, true);
          linear_solver(&sigma_, A, rhs_, 1e-6, 50);
          sigma.ReInit(Nelem * COORD_DIM);
          for (Long i = 0; i < Nelem*COORD_DIM; i++) {// Set sigma
            for (Long k = 0; k < Nnode; k++) {
              sigma[i][k] = sigma_[i*Nnode+k] - sigma0[i][k];
            }
          }
        }

        S.quadrature_FxU.Eval(U, S.GetElem(), sigma, S.Stokes_FxU);
        { // Write VTU
          VTUData vtu_sigma;
          vtu_sigma.AddElems(S.elements, sigma, ORDER);
          vtu_sigma.WriteVTK("sphere-sigma0", comm);

          VTUData vtu_U;
          vtu_U.AddElems(S.elements, U, ORDER);
          vtu_U.WriteVTK("sphere-U0", comm);
        }
      }

      { // Tornberg
        Vector<DensityBasis> rhs;
        BIOpL(rhs, U);

        Vector<DensityBasis> sigma;
        { // Set sigma
          Long Nnode = DensityBasis::Size();
          Long Nelem = S.GetElem().NElem();

          typename sctl::ParallelSolver<Real>::ParallelOp A = [&S,&BIOp_half_S_D](sctl::Vector<Real>* Ax, const sctl::Vector<Real>& x) {
            Long Nnode = DensityBasis::Size();
            Long Nelem = S.GetElem().NElem();
            Ax->ReInit(Nelem*COORD_DIM*Nnode);

            Vector<DensityBasis> x_(Nelem*COORD_DIM), Ax_(Nelem*COORD_DIM);
            for (Long i = 0; i < Nelem*COORD_DIM; i++) { // Set x_
              for (Long k = 0; k < Nnode; k++) {
                x_[i][k] = x[i*Nnode+k];
              }
            }
            BIOp_half_S_D(Ax_, x_);
            for (Long i = 0; i < Nelem*COORD_DIM; i++) { // Set Ax
              for (Long k = 0; k < Nnode; k++) {
                (*Ax)[i*Nnode+k] = Ax_[i][k];
              }
            }
          };

          Vector<Real> sigma_(Nelem*COORD_DIM*Nnode), rhs_(Nelem*COORD_DIM*Nnode);
          for (Long i = 0; i < Nelem*COORD_DIM; i++) {// Set rhs_
            for (Long k = 0; k < Nnode; k++) {
              rhs_[i*Nnode+k] = rhs[i][k];
            }
          }
          sigma_ = 0;

          ParallelSolver<Real> linear_solver(comm, true);
          linear_solver(&sigma_, A, rhs_, 1e-6, 50);
          sigma.ReInit(Nelem * COORD_DIM);
          for (Long i = 0; i < Nelem*COORD_DIM; i++) {// Set sigma
            for (Long k = 0; k < Nnode; k++) {
              sigma[i][k] = sigma_[i*Nnode+k];
            }
          }
        }

        Vector<PotentialBasis> U1;
        BIOp_half_S_D(U1, sigma);
        { // Write VTU
          VTUData vtu_sigma;
          vtu_sigma.AddElems(S.elements, sigma, ORDER);
          vtu_sigma.WriteVTK("sphere-sigma1", comm);

          VTUData vtu_U;
          vtu_U.AddElems(S.elements, U1, ORDER);
          vtu_U.WriteVTK("sphere-U1", comm);
        }
      }

      Profile::print(&comm);
    }

  private:

    template <class FnBasis> void SurfInteg(Vector<Real>& I, const Vector<FnBasis>& f) {
      static_assert(std::is_same<FnBasis,CoordBasis>::value, "FnBasis is different from CoordBasis");
      const Long Nelem = elements.NElem();
      const Long dof = f.Dim() / Nelem;
      SCTL_ASSERT(f.Dim() == Nelem * dof);

      auto nodes = FnBasis::Nodes();
      auto quad_wts = FnBasis::QuadWts();
      const Long Nnodes = FnBasis::Size();
      auto EvalOp = CoordBasis::SetupEval(nodes);

      Vector<CoordBasis> dX;
      const auto& X = elements.ElemVector();
      SCTL_ASSERT(X.Dim() == Nelem * COORD_DIM);
      CoordBasis::Grad(dX, X);

      Matrix<Real> I_(Nelem, dof);
      for (Long i = 0; i < Nelem; i++) {
        for (Long k = 0; k < dof; k++) {
          I_[i][k] = 0;
        }
        for (Long j = 0; j < Nnodes; j++) {
          Real dA = 0;
          StaticArray<Real,COORD_DIM> Xn;
          Xn[0] = dX[i*COORD_DIM*2+2][j] * dX[i*COORD_DIM*2+5][j] - dX[i*COORD_DIM*2+3][j] * dX[i*COORD_DIM*2+4][j];
          Xn[1] = dX[i*COORD_DIM*2+4][j] * dX[i*COORD_DIM*2+1][j] - dX[i*COORD_DIM*2+5][j] * dX[i*COORD_DIM*2+0][j];
          Xn[2] = dX[i*COORD_DIM*2+0][j] * dX[i*COORD_DIM*2+3][j] - dX[i*COORD_DIM*2+1][j] * dX[i*COORD_DIM*2+2][j];
          dA += sqrt<Real>(Xn[0]*Xn[0] + Xn[1]*Xn[1] + Xn[2]*Xn[2]) * quad_wts[j];
          for (Long k = 0; k < dof; k++) {
            I_[i][k] += dA * f[i*dof+k][j];
          }
        }
      }

      Long Ns = elem_cnt.Dim();
      if (I.Dim() != Ns * dof) I.ReInit(Ns * dof);
      I = 0;
      Long elem_itr = 0;
      for (Long i = 0; i < Ns; i++) {
        for (Long j = 0; j < elem_cnt[i]; j++) {
          for (Long k = 0; k < dof; k++) {
            I[i*dof+k] += I_[elem_itr][k];
          }
          elem_itr++;
        }
      }
    }

    void InitSpheres(const Vector<Real> X, const Vector<Real>& R){
      SCTL_ASSERT(X.Dim() == R.Dim() * COORD_DIM);
      Long N = R.Dim();
      elements.ReInit(2*COORD_DIM*N);
      auto nodes = ElemLst::CoordBasis::Nodes();
      for (Long l = 0; l < N; l++) {
        for (Integer i = 0; i < COORD_DIM; i++) {
          for (Integer j = 0; j < 2; j++) {
            for (int k = 0; k < ElemLst::CoordBasis::Size(); k++) {
              Real coord[COORD_DIM];
              coord[(i+0)%COORD_DIM] = (j ? -1.0 : 1.0);
              coord[(i+1)%COORD_DIM] = 2.0 * nodes[j?1:0][k] - 1.0;
              coord[(i+2)%COORD_DIM] = 2.0 * nodes[j?0:1][k] - 1.0;
              Real R0 = sqrt<Real>(coord[0]*coord[0] + coord[1]*coord[1] + coord[2]*coord[2]);

              elements((l*COORD_DIM+i)*2+j,0)[k] = X[l*COORD_DIM+0] + R[l] * coord[0] / R0;
              elements((l*COORD_DIM+i)*2+j,1)[k] = X[l*COORD_DIM+1] + R[l] * coord[1] / R0;
              elements((l*COORD_DIM+i)*2+j,2)[k] = X[l*COORD_DIM+2] + R[l] * coord[2] / R0;
            }
          }
        }
      }
      elem_cnt.ReInit(N);
      elem_cnt = 6;
    }

    GenericKernel<Stokes3D_DxU> Stokes_DxU;
    GenericKernel<Stokes3D_FxU> Stokes_FxU;
    GenericKernel<Stokes3D_FxT> Stokes_FxT;

    Quadrature<Real> quadrature_DxU;
    Quadrature<Real> quadrature_FxU;
    Quadrature<Real> quadrature_FxT;

    ElemLst elements;
    Vector<Long> elem_cnt;
};

}  // end namespace

#endif  //_SCTL_BOUNDARY_QUADRATURE_HPP_