void solverloop(int start, int end){
int i, j;
int indx, indx_up, indx_lft;
int indx_rght, indx_up, indx_dwn;
double phi,dphi_dt,dmu_dt;
double drv_frce, alln_chn;
double Gamma, kai;
double dc_dx, dc_dy, V_gradC;
#ifdef ANISO
grad_phi(1, dphi_now);
#endif
for (i=start; i < end; i++) {
#ifdef ANISO
grad_phi(i+1, dphi_next);
#endif
for (j=1; j < (MESHX-1); j++){
indx = i*MESHX + j;
indx_lft = indx - 1;
indx_rght = indx + 1;
indx_up = indx - MESHX;
indx_dwn = indx + MESHX;
phi = phi_old[indx];
#ifdef ISO
Gamma = 2*G*lap_phi[indx];
#endif
#ifdef ANISO
Gamma = div_phi(j);
#endif
drv_frce = (mu_old[indx] - Mu)*(K-1)*(mu_old[indx])*6*phi*(1-phi);
alln_chn = E*Gamma - (G/E)*18.0*(phi)*(1.0-phi)*(1.0-2.0*phi);
dp_dt = (alln_chn + drv_frce)/(tau*E);
phi_new[z] = phi + deltat*dphi_dt;
#ifdef ANISO
dc_dx = (conc[indx_rght] - conc[indx_lft])*0.5*inv_deltax;
dc_dy = (conc[indx_dwn] - conc[indx_up] )*0.5*inv_deltax;
V_gradC = u_old[indx]*dc_dx + v_old[indx]*dc_dy;
#endif
#ifdef ISO
V_gradC = 0.0;
#endif
dmu_dt = Mob*lap_mu[indx] - V_gradC - (K-1)*mu_old[indx]*6*phi*(1-phi)*dphi_dt;
kai = 1+(K-1)*phi*phi*(3-2*phi);
mu_new[indx] = mu_old[indx] + deltat*dmu_dt/kai;
}
#ifdef ANISO
fnupdate();
#endif
}
}
void solverloop(){
int i, j, z;
double p,dp_dt,dmu_dt;
double drv_frce, alln_chn;
double Gamma, kai;
double dc_dx, dc_dy, V_gradC = 0.0;
#ifdef ANISO
grad_phi(1, dphi_now);
#endif
for (i=start; i <= end; i++) {
#ifdef ANISO
grad_phi(i+1, dphi_next);
#endif
for (j=1; j < (MESHX-1); j++){
z = i*MESHX + j;
p = phi_old[z];
#ifdef ISO
Gamma = 2*G*lap_phi[z];
#endif
#ifdef ANISO
Gamma = div_phi(j);
#endif
drv_frce = (mu_old[z] - Mu)*(K-1)*(mu_old[z])*6*p*(1-p);
alln_chn = E*Gamma - (G/E)*18.0*(p)*(1.0-p)*(1.0-2.0*p);
dp_dt = (alln_chn + drv_frce)/(tau*E);
phi_new[z] = p + deltat*dp_dt;
#ifdef FLUID
dc_dx = (conc[z+1]-conc[z-1])*0.5*inv_deltax;
dc_dy = (conc[z+MESHX]-conc[z-MESHX])*0.5*inv_deltax;
V_gradC = u_old[z]*dc_dx + v_old[z]*dc_dy;
#endif
dmu_dt = Mob*lap_mu[z] - V_gradC - (K-1)*mu_old[z]*6*p*(1-p)*dp_dt;
kai = 1+(K-1)*p*p*(3-2*p);
mu_new[z] = mu_old[z] + deltat*dmu_dt/kai;
}
#ifdef ANISO
fnupdate();
#endif
}
}
void solve_with_Cpp(MultiFab& soln, MultiFab& gphi, Real a, Real b, MultiFab& alpha,
PArray<MultiFab>& beta, MultiFab& rhs, const BoxArray& bs, const Geometry& geom)
{
BL_PROFILE("solve_with_Cpp()");
BndryData bd(bs, 1, geom);
set_boundary(bd, rhs, 0);
ABecLaplacian abec_operator(bd, dx);
abec_operator.setScalars(a, b);
abec_operator.setCoefficients(alpha, beta);
MultiGrid mg(abec_operator);
mg.setVerbose(verbose);
mg.solve(soln, rhs, tolerance_rel, tolerance_abs);
PArray<MultiFab> grad_phi(BL_SPACEDIM, PArrayManage);
for (int n = 0; n < BL_SPACEDIM; ++n)
grad_phi.set(n, new MultiFab(BoxArray(soln.boxArray()).surroundingNodes(n), 1, 0));
#if (BL_SPACEDIM == 2)
abec_operator.compFlux(grad_phi[0],grad_phi[1],soln);
#elif (BL_SPACEDIM == 3)
abec_operator.compFlux(grad_phi[0],grad_phi[1],grad_phi[2],soln);
#endif
// Average edge-centered gradients to cell centers.
BoxLib::average_face_to_cellcenter(gphi, grad_phi, geom);
}
void solve_with_F90(MultiFab& soln, MultiFab& gphi, Real a, Real b, MultiFab& alpha,
PArray<MultiFab>& beta, MultiFab& rhs, const BoxArray& bs, const Geometry& geom)
{
BL_PROFILE("solve_with_F90()");
FMultiGrid fmg(geom);
int mg_bc[2*BL_SPACEDIM];
if (bc_type == Periodic) {
// Define the type of boundary conditions to be periodic
for ( int n = 0; n < BL_SPACEDIM; ++n ) {
mg_bc[2*n + 0] = MGT_BC_PER;
mg_bc[2*n + 1] = MGT_BC_PER;
}
}
else if (bc_type == Neumann) {
// Define the type of boundary conditions to be Neumann
for ( int n = 0; n < BL_SPACEDIM; ++n ) {
mg_bc[2*n + 0] = MGT_BC_NEU;
mg_bc[2*n + 1] = MGT_BC_NEU;
}
}
else if (bc_type == Dirichlet) {
// Define the type of boundary conditions to be Dirichlet
for ( int n = 0; n < BL_SPACEDIM; ++n ) {
mg_bc[2*n + 0] = MGT_BC_DIR;
mg_bc[2*n + 1] = MGT_BC_DIR;
}
}
fmg.set_bc(mg_bc);
fmg.set_maxorder(maxorder);
fmg.set_scalars(a, b);
fmg.set_coefficients(alpha, beta);
int always_use_bnorm = 0;
int need_grad_phi = 1;
fmg.solve(soln, rhs, tolerance_rel, tolerance_abs, always_use_bnorm, need_grad_phi);
PArray<MultiFab> grad_phi(BL_SPACEDIM, PArrayManage);
for (int n = 0; n < BL_SPACEDIM; ++n)
grad_phi.set(n, new MultiFab(BoxArray(soln.boxArray()).surroundingNodes(n), 1, 0));
fmg.get_fluxes(grad_phi);
// Average edge-centered gradients to cell centers.
BoxLib::average_face_to_cellcenter(gphi, grad_phi, geom);
}
KOKKOS_INLINE_FUNCTION
void FEInterpolation<EvalT, Traits>::operator () (const int i) const
{
for (PHX::index_size_type qp = 0; qp < num_qp; ++qp) {
val_qp(i,qp) = 0.0;
for (PHX::index_size_type dim = 0; dim < num_dim; ++dim)
val_grad_qp(i,qp,dim) = 0.0;
// Sum nodal contributions to qp
for (PHX::index_size_type node = 0; node < num_nodes; ++node) {
val_qp(i,qp) += phi(qp, node) * val_node(i,node);
for (PHX::index_size_type dim = 0; dim < num_dim; ++dim)
val_grad_qp(i,qp,dim) += grad_phi(qp, node, dim) * val_node(i,node);
}
}
}
//**********************************************************************
PHX_EVALUATE_FIELDS(FEInterpolation,cell_data)
{
std::vector<Element_Linear2D>::iterator cell_it = cell_data.begin;
// Loop over number of cells
for (std::size_t cell = 0; cell < cell_data.num_cells; ++cell) {
const shards::Array<double,shards::NaturalOrder,QuadPoint,Node>& phi =
cell_it->basisFunctions();
const shards::Array<double,shards::NaturalOrder,QuadPoint,Node,Dim>&
grad_phi = cell_it->basisFunctionGradientsRealSpace();
// Loop over quad points of cell
for (int qp = 0; qp < num_qp; ++qp) {
val_qp(cell,qp) = 0.0;
for (int dim = 0; dim < num_dim; ++dim)
val_grad_qp(cell,qp,dim) = 0.0;
// Sum nodal contributions to qp
for (int node = 0; node < num_nodes; ++node) {
val_qp(cell,qp) += phi(qp,node) * val_node(cell,node);
for (int dim = 0; dim < num_dim; ++dim)
val_grad_qp(cell,qp,dim) +=
grad_phi(qp,node,dim) * val_node(cell,node);
}
}
++cell_it;
}
// std::cout << "FEINterpolation: val_node" << std::endl;
// val_node.print(std::cout,true);
// std::cout << "FEINterpolation: val_qp" << std::endl;
// val_qp.print(std::cout,true);
// std::cout << "FEINterpolation: val_grad_qp" << std::endl;
// val_grad_qp.print(std::cout,true);
}
void CellwiseDataGradient<DIM>::SetupGradients(AbstractCellPopulation<DIM>& rCellPopulation, const std::string& rItemName)
{
MeshBasedCellPopulation<DIM>* pCellPopulation = static_cast<MeshBasedCellPopulation<DIM>*>(&(rCellPopulation));
TetrahedralMesh<DIM,DIM>& r_mesh = pCellPopulation->rGetMesh();
// Initialise gradients size
unsigned num_nodes = pCellPopulation->GetNumNodes();
mGradients.resize(num_nodes, zero_vector<double>(DIM));
// The constant gradients at each element
std::vector<c_vector<double, DIM> > gradients_on_elements;
unsigned num_elements = r_mesh.GetNumElements();
gradients_on_elements.resize(num_elements, zero_vector<double>(DIM));
// The number of elements containing a given node (excl ghost elements)
std::vector<unsigned> num_real_elems_for_node(num_nodes, 0);
for (unsigned elem_index=0; elem_index<num_elements; elem_index++)
{
Element<DIM,DIM>& r_elem = *(r_mesh.GetElement(elem_index));
// Calculate the basis functions at any point (eg zero) in the element
c_matrix<double, DIM, DIM> jacobian, inverse_jacobian;
double jacobian_det;
r_mesh.GetInverseJacobianForElement(elem_index, jacobian, jacobian_det, inverse_jacobian);
const ChastePoint<DIM> zero_point;
c_matrix<double, DIM, DIM+1> grad_phi;
LinearBasisFunction<DIM>::ComputeTransformedBasisFunctionDerivatives(zero_point, inverse_jacobian, grad_phi);
bool is_ghost_element = false;
for (unsigned node_index=0; node_index<DIM+1; node_index++)
{
unsigned node_global_index = r_elem.GetNodeGlobalIndex(node_index);
// This code is commented because CellData can't deal with ghost nodes see #1975
assert(pCellPopulation->IsGhostNode(node_global_index) == false);
//// Check whether ghost element
//if (pCellPopulation->IsGhostNode(node_global_index) == true)
//{
// is_ghost_element = true;
// break;
//}
// If no ghost element, get PDE solution
CellPtr p_cell = pCellPopulation->GetCellUsingLocationIndex(node_global_index);
double pde_solution = p_cell->GetCellData()->GetItem(rItemName);
// Interpolate gradient
for (unsigned i=0; i<DIM; i++)
{
gradients_on_elements[elem_index](i) += pde_solution* grad_phi(i, node_index);
}
}
// Add gradient at element to gradient at node
if (!is_ghost_element)
{
for (unsigned node_index=0; node_index<DIM+1; node_index++)
{
unsigned node_global_index = r_elem.GetNodeGlobalIndex(node_index);
mGradients[node_global_index] += gradients_on_elements[elem_index];
num_real_elems_for_node[node_global_index]++;
}
}
}
// Divide to obtain average gradient
for (typename AbstractCellPopulation<DIM>::Iterator cell_iter = pCellPopulation->Begin();
cell_iter != pCellPopulation->End();
++cell_iter)
{
unsigned node_global_index = pCellPopulation->GetLocationIndexUsingCell(*cell_iter);
if (!num_real_elems_for_node[node_global_index] > 0)
{
NEVER_REACHED;
// This code is commented because CellwiseData Can't deal with ghost nodes so won't ever come into this statement see #1975
//// The node is a real node which is not in any real element
//// but should be connected to some cells (if more than one cell in mesh)
//Node<DIM>& this_node = *(pCellPopulation->GetNodeCorrespondingToCell(*cell_iter));
//
//mGradients[node_global_index] = zero_vector<double>(DIM);
//unsigned num_real_adjacent_nodes = 0;
//
//// Get all the adjacent nodes which correspond to real cells
//std::set<Node<DIM>*> real_adjacent_nodes;
//real_adjacent_nodes.clear();
//
//// First loop over containing elements
//for (typename Node<DIM>::ContainingElementIterator element_iter = this_node.ContainingElementsBegin();
// element_iter != this_node.ContainingElementsEnd();
// ++element_iter)
//{
// // Then loop over nodes therein
// Element<DIM,DIM>& r_adjacent_elem = *(r_mesh.GetElement(*element_iter));
// for (unsigned local_node_index=0; local_node_index<DIM+1; local_node_index++)
// {
// unsigned adjacent_node_global_index = r_adjacent_elem.GetNodeGlobalIndex(local_node_index);
//
// // If not a ghost node and not the node we started with
// if ( !(pCellPopulation->IsGhostNode(adjacent_node_global_index))
// && adjacent_node_global_index != node_global_index )
// {
//
// // Calculate the contribution of gradient from this node
// Node<DIM>& adjacent_node = *(r_mesh.GetNode(adjacent_node_global_index));
//
// double this_cell_concentration = CellwiseData<DIM>::Instance()->GetValue(*cell_iter, 0);
// CellPtr p_adjacent_cell = pCellPopulation->GetCellUsingLocationIndex(adjacent_node_global_index);
// double adjacent_cell_concentration = CellwiseData<DIM>::Instance()->GetValue(p_adjacent_cell, 0);
//
// c_vector<double, DIM> gradient_contribution = zero_vector<double>(DIM);
//
// if (fabs(this_cell_concentration-adjacent_cell_concentration) > 100*DBL_EPSILON)
// {
// c_vector<double, DIM> edge_vector = r_mesh.GetVectorFromAtoB(this_node.rGetLocation(), adjacent_node.rGetLocation());
// double norm_edge_vector = norm_2(edge_vector);
// gradient_contribution = edge_vector
// * (adjacent_cell_concentration - this_cell_concentration)
// / (norm_edge_vector * norm_edge_vector);
// }
//
// mGradients[node_global_index] += gradient_contribution;
// num_real_adjacent_nodes++;
// }
// }
//}
//mGradients[node_global_index] /= num_real_adjacent_nodes;
}
else
{
mGradients[node_global_index] /= num_real_elems_for_node[node_global_index];
}
}
}
void solve_with_HPGMG(MultiFab& soln, MultiFab& gphi, Real a, Real b, MultiFab& alpha, PArray<MultiFab>& beta,
MultiFab& beta_cc, MultiFab& rhs, const BoxArray& bs, const Geometry& geom, int n_cell)
{
BndryData bd(bs, 1, geom);
set_boundary(bd, rhs, 0);
ABecLaplacian abec_operator(bd, dx);
abec_operator.setScalars(a, b);
abec_operator.setCoefficients(alpha, beta);
int minCoarseDim;
if (domain_boundary_condition == BC_PERIODIC)
{
minCoarseDim = 2; // avoid problems with black box calculation of D^{-1} for poisson with periodic BC's on a 1^3 grid
}
else
{
minCoarseDim = 1; // assumes you can drop order on the boundaries
}
level_type level_h;
mg_type MG_h;
int numVectors = 12;
int my_rank = 0, num_ranks = 1;
#ifdef BL_USE_MPI
MPI_Comm_size (MPI_COMM_WORLD, &num_ranks);
MPI_Comm_rank (MPI_COMM_WORLD, &my_rank);
#endif /* BL_USE_MPI */
const double h0 = dx[0];
// Create the geometric structure of the HPGMG grid using the RHS MultiFab as
// a template. This doesn't copy any actual data.
CreateHPGMGLevel(&level_h, rhs, n_cell, max_grid_size, my_rank, num_ranks, domain_boundary_condition, numVectors, h0);
// Set up the coefficients for the linear operator L.
SetupHPGMGCoefficients(a, b, alpha, beta_cc, &level_h);
// Now that the HPGMG grid is built, populate it with RHS data.
ConvertToHPGMGLevel(rhs, n_cell, max_grid_size, &level_h, VECTOR_F);
#ifdef USE_HELMHOLTZ
if (ParallelDescriptor::IOProcessor()) {
std::cout << "Creating Helmholtz (a=" << a << ", b=" << b << ") test problem" << std::endl;;
}
#else
if (ParallelDescriptor::IOProcessor()) {
std::cout << "Creating Poisson (a=" << a << ", b=" << b << ") test problem" << std::endl;;
}
#endif /* USE_HELMHOLTZ */
if (level_h.boundary_condition.type == BC_PERIODIC)
{
double average_value_of_f = mean (&level_h, VECTOR_F);
if (average_value_of_f != 0.0)
{
if (ParallelDescriptor::IOProcessor())
{
std::cerr << "WARNING: Periodic boundary conditions, but f does not sum to zero... mean(f)=" << average_value_of_f << std::endl;
}
//shift_vector(&level_h,VECTOR_F,VECTOR_F,-average_value_of_f);
}
}
//- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
rebuild_operator(&level_h,NULL,a,b); // i.e. calculate Dinv and lambda_max
MGBuild(&MG_h,&level_h,a,b,minCoarseDim,ParallelDescriptor::Communicator()); // build the Multigrid Hierarchy
//- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
if (ParallelDescriptor::IOProcessor())
std::cout << std::endl << std::endl << "===== STARTING SOLVE =====" << std::endl << std::flush;
MGResetTimers (&MG_h);
zero_vector (MG_h.levels[0], VECTOR_U);
#ifdef USE_FCYCLES
FMGSolve (&MG_h, 0, VECTOR_U, VECTOR_F, a, b, tolerance_abs, tolerance_rel);
#else
MGSolve (&MG_h, 0, VECTOR_U, VECTOR_F, a, b, tolerance_abs, tolerance_rel);
#endif /* USE_FCYCLES */
MGPrintTiming (&MG_h, 0); // don't include the error check in the timing results
//- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
if (ParallelDescriptor::IOProcessor())
std::cout << std::endl << std::endl << "===== Performing Richardson error analysis ==========================" << std::endl;
// solve A^h u^h = f^h
// solve A^2h u^2h = f^2h
// solve A^4h u^4h = f^4h
// error analysis...
MGResetTimers(&MG_h);
const double dtol = tolerance_abs;
const double rtol = tolerance_rel;
int l;for(l=0;l<3;l++){
if(l>0)restriction(MG_h.levels[l],VECTOR_F,MG_h.levels[l-1],VECTOR_F,RESTRICT_CELL);
zero_vector(MG_h.levels[l],VECTOR_U);
#ifdef USE_FCYCLES
FMGSolve(&MG_h,l,VECTOR_U,VECTOR_F,a,b,dtol,rtol);
#else
MGSolve(&MG_h,l,VECTOR_U,VECTOR_F,a,b,dtol,rtol);
#endif
}
richardson_error(&MG_h,0,VECTOR_U);
// Now convert solution from HPGMG back to rhs MultiFab.
ConvertFromHPGMGLevel(soln, &level_h, VECTOR_U);
const double norm_from_HPGMG = norm(&level_h, VECTOR_U);
const double mean_from_HPGMG = mean(&level_h, VECTOR_U);
const Real norm0 = soln.norm0();
const Real norm2 = soln.norm2();
if (ParallelDescriptor::IOProcessor()) {
std::cout << "mean from HPGMG: " << mean_from_HPGMG << std::endl;
std::cout << "norm from HPGMG: " << norm_from_HPGMG << std::endl;
std::cout << "norm0 of RHS copied to MF: " << norm0 << std::endl;
std::cout << "norm2 of RHS copied to MF: " << norm2 << std::endl;
}
// Write the MF to disk for comparison with the in-house solver
if (plot_soln)
{
writePlotFile("SOLN-HPGMG", soln, geom);
}
//- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
MGDestroy(&MG_h);
destroy_level(&level_h);
//- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
PArray<MultiFab> grad_phi(BL_SPACEDIM, PArrayManage);
for (int n = 0; n < BL_SPACEDIM; ++n)
grad_phi.set(n, new MultiFab(BoxArray(soln.boxArray()).surroundingNodes(n), 1, 0));
#if (BL_SPACEDIM == 2)
abec_operator.compFlux(grad_phi[0],grad_phi[1],soln);
#elif (BL_SPACEDIM == 3)
abec_operator.compFlux(grad_phi[0],grad_phi[1],grad_phi[2],soln);
#endif
// Average edge-centered gradients to cell centers.
BoxLib::average_face_to_cellcenter(gphi, grad_phi, geom);
}
double AbstractFunctionalCalculator<ELEMENT_DIM, SPACE_DIM, PROBLEM_DIM>::CalculateOnElement(Element<ELEMENT_DIM, SPACE_DIM>& rElement)
{
double result_on_element = 0;
// Third order quadrature. Note that the functional may be non-polynomial (see documentation of class).
GaussianQuadratureRule<ELEMENT_DIM> quad_rule(3);
/// NOTE: This assumes that the Jacobian is constant on an element, ie
/// no curvilinear bases were used for position
double jacobian_determinant;
c_matrix<double, SPACE_DIM, ELEMENT_DIM> jacobian;
c_matrix<double, ELEMENT_DIM, SPACE_DIM> inverse_jacobian;
rElement.CalculateInverseJacobian(jacobian, jacobian_determinant, inverse_jacobian);
const unsigned num_nodes = rElement.GetNumNodes();
// Loop over Gauss points
for (unsigned quad_index=0; quad_index < quad_rule.GetNumQuadPoints(); quad_index++)
{
const ChastePoint<ELEMENT_DIM>& quad_point = quad_rule.rGetQuadPoint(quad_index);
c_vector<double, ELEMENT_DIM+1> phi;
LinearBasisFunction<ELEMENT_DIM>::ComputeBasisFunctions(quad_point, phi);
c_matrix<double, ELEMENT_DIM, ELEMENT_DIM+1> grad_phi;
LinearBasisFunction<ELEMENT_DIM>::ComputeTransformedBasisFunctionDerivatives(quad_point, inverse_jacobian, grad_phi);
// Location of the Gauss point in the original element will be stored in x
ChastePoint<SPACE_DIM> x(0,0,0);
c_vector<double,PROBLEM_DIM> u = zero_vector<double>(PROBLEM_DIM);
c_matrix<double,PROBLEM_DIM,SPACE_DIM> grad_u = zero_matrix<double>(PROBLEM_DIM,SPACE_DIM);
for (unsigned i=0; i<num_nodes; i++)
{
const c_vector<double, SPACE_DIM>& r_node_loc = rElement.GetNode(i)->rGetLocation();
// Interpolate x
x.rGetLocation() += phi(i)*r_node_loc;
// Interpolate u and grad u
unsigned node_global_index = rElement.GetNodeGlobalIndex(i);
for (unsigned index_of_unknown=0; index_of_unknown<PROBLEM_DIM; index_of_unknown++)
{
// NOTE - following assumes that, if say there are two unknowns u and v, they
// are stored in the current solution vector as
// [U1 V1 U2 V2 ... U_n V_n]
unsigned index_into_vec = PROBLEM_DIM*node_global_index + index_of_unknown;
double u_at_node = mSolutionReplicated[index_into_vec];
u(index_of_unknown) += phi(i)*u_at_node;
for (unsigned j=0; j<SPACE_DIM; j++)
{
grad_u(index_of_unknown,j) += grad_phi(j,i)*u_at_node;
}
}
}
double wJ = jacobian_determinant * quad_rule.GetWeight(quad_index);
result_on_element += GetIntegrand(x, u, grad_u) * wJ;
}
return result_on_element;
}