test_02.cpp

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#include "Intrepid_FieldContainer.hpp"
#include "Intrepid_HGRAD_HEX_Cn_FEM.hpp"
#include "Intrepid_PointTools.hpp"
#include "Intrepid_DefaultCubatureFactory.hpp"
#include "Intrepid_RealSpaceTools.hpp"
#include "Intrepid_ArrayTools.hpp"
#include "Intrepid_FunctionSpaceTools.hpp"
#include "Intrepid_CellTools.hpp"
#include "Teuchos_oblackholestream.hpp"
#include "Teuchos_RCP.hpp"
#include "Teuchos_GlobalMPISession.hpp"
#include "Teuchos_SerialDenseMatrix.hpp"
#include "Teuchos_SerialDenseVector.hpp"
#include "Teuchos_LAPACK.hpp"
 
using namespace std;
using namespace Intrepid;
 
void rhsFunc(FieldContainer<double> &, const FieldContainer<double> &, int, int, int);
void neumann(FieldContainer<double>       & ,
             const FieldContainer<double> & ,
             const FieldContainer<double> & ,
             const shards::CellTopology   & ,
             int, int, int, int);
void u_exact(FieldContainer<double> &, const FieldContainer<double> &, int, int, int);
 
void rhsFunc(FieldContainer<double> & result,
             const FieldContainer<double> & points,
             int xd,
             int yd,
             int zd) {
 
  int x = 0, y = 1, z = 2;
 
  // second x-derivatives of u
  if (xd > 1) {
    for (int cell=0; cell<result.dimension(0); cell++) {
      for (int pt=0; pt<result.dimension(1); pt++) {
        result(cell,pt) = - xd*(xd-1)*std::pow(points(cell,pt,x), xd-2) *
                            std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
      }
    }
  }
 
  // second y-derivatives of u
  if (yd > 1) {
    for (int cell=0; cell<result.dimension(0); cell++) {
      for (int pt=0; pt<result.dimension(1); pt++) {
        result(cell,pt) -=  yd*(yd-1)*std::pow(points(cell,pt,y), yd-2) *
                            std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,z), zd);
      }
    }
  }
 
  // second z-derivatives of u
  if (zd > 1) {
    for (int cell=0; cell<result.dimension(0); cell++) {
      for (int pt=0; pt<result.dimension(1); pt++) {
        result(cell,pt) -=  zd*(zd-1)*std::pow(points(cell,pt,z), zd-2) *
                            std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd);
      }
    }
  }
 
  // add u
  for (int cell=0; cell<result.dimension(0); cell++) {
    for (int pt=0; pt<result.dimension(1); pt++) {
      result(cell,pt) +=  std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
    }
  }
 
}
 
 
void neumann(FieldContainer<double>       & result,
             const FieldContainer<double> & points,
             const FieldContainer<double> & jacs,
             const shards::CellTopology   & parentCell,
             int sideOrdinal, int xd, int yd, int zd) {
 
  int x = 0, y = 1, z = 2;
 
  int numCells  = result.dimension(0);
  int numPoints = result.dimension(1);
 
  FieldContainer<double> grad_u(numCells, numPoints, 3);
  FieldContainer<double> side_normals(numCells, numPoints, 3);
  FieldContainer<double> normal_lengths(numCells, numPoints);
 
  // first x-derivatives of u
  if (xd > 0) {
    for (int cell=0; cell<numCells; cell++) {
      for (int pt=0; pt<numPoints; pt++) {
        grad_u(cell,pt,x) = xd*std::pow(points(cell,pt,x), xd-1) *
                            std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
      }
    }
  }
 
  // first y-derivatives of u
  if (yd > 0) {
    for (int cell=0; cell<numCells; cell++) {
      for (int pt=0; pt<numPoints; pt++) {
        grad_u(cell,pt,y) = yd*std::pow(points(cell,pt,y), yd-1) *
                            std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,z), zd);
      }
    }
  }
 
  // first z-derivatives of u
  if (zd > 0) {
    for (int cell=0; cell<numCells; cell++) {
      for (int pt=0; pt<numPoints; pt++) {
        grad_u(cell,pt,z) = zd*std::pow(points(cell,pt,z), zd-1) *
                            std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd);
      }
    }
  }
  
  CellTools<double>::getPhysicalSideNormals(side_normals, jacs, sideOrdinal, parentCell);
 
  // scale normals
  RealSpaceTools<double>::vectorNorm(normal_lengths, side_normals, NORM_TWO);
  FunctionSpaceTools::scalarMultiplyDataData<double>(side_normals, normal_lengths, side_normals, true); 
 
  FunctionSpaceTools::dotMultiplyDataData<double>(result, grad_u, side_normals);
 
}
 
void u_exact(FieldContainer<double> & result, const FieldContainer<double> & points, int xd, int yd, int zd) {
  int x = 0, y = 1, z = 2;
  for (int cell=0; cell<result.dimension(0); cell++) {
    for (int pt=0; pt<result.dimension(1); pt++) {
      result(cell,pt) = std::pow(points(pt,x), xd)*std::pow(points(pt,y), yd)*std::pow(points(pt,z), zd);
    }
  }
}
 
 
 
 
int main(int argc, char *argv[]) {
 
  Teuchos::GlobalMPISession mpiSession(&argc, &argv);
 
  // This little trick lets us print to std::cout only if
  // a (dummy) command-line argument is provided.
  int iprint     = argc - 1;
  Teuchos::RCP<std::ostream> outStream;
  Teuchos::oblackholestream bhs; // outputs nothing
  if (iprint > 0)
    outStream = Teuchos::rcp(&std::cout, false);
  else
    outStream = Teuchos::rcp(&bhs, false);
 
  // Save the format state of the original std::cout.
  Teuchos::oblackholestream oldFormatState;
  oldFormatState.copyfmt(std::cout);
 
  *outStream \
    << "===============================================================================\n" \
    << "|                                                                             |\n" \
    << "|                    Unit Test (Basis_HGRAD_HEX_Cn_FEM)                       |\n" \
    << "|                                                                             |\n" \
    << "|     1) Patch test involving mass and stiffness matrices,                    |\n" \
    << "|        for the Neumann problem on a physical parallelepiped                 |\n" \
    << "|        AND a reference hex Omega with boundary Gamma.                       |\n" \
    << "|                                                                             |\n" \
    << "|        - div (grad u) + u = f  in Omega,  (grad u) . n = g  on Gamma        |\n" \
    << "|                                                                             |\n" \
    << "|        For a generic parallelepiped, the basis recovers a complete          |\n" \
    << "|        polynomial space of order 2. On a (scaled and/or translated)         |\n" \
    << "|        reference hex, the basis recovers a complete tensor product          |\n" \
    << "|        space of order 1 (i.e. incl. cross terms, e.g. x^2*y^2*z^2).         |\n" \
    << "|                                                                             |\n" \
    << "|  Questions? Contact  Pavel Bochev  (pbboche@sandia.gov),                    |\n" \
    << "|                      Denis Ridzal  (dridzal@sandia.gov),                    |\n" \
    << "|                      Kara Peterson (kjpeter@sandia.gov).                    |\n" \
    << "|                                                                             |\n" \
    << "|  Intrepid's website: http://trilinos.sandia.gov/packages/intrepid           |\n" \
    << "|  Trilinos website:   http://trilinos.sandia.gov                             |\n" \
    << "|                                                                             |\n" \
    << "===============================================================================\n"\
    << "| TEST 1: Patch test                                                          |\n"\
    << "===============================================================================\n";
 
  
  int errorFlag = 0;
 
  outStream -> precision(16);
 
 
  try {
 
    int max_order = 3;                                                                    // max total order of polynomial solution
    DefaultCubatureFactory<double>  cubFactory;                                           // create factory
    shards::CellTopology cell(shards::getCellTopologyData< shards::Hexahedron<> >());     // create parent cell topology
    shards::CellTopology side(shards::getCellTopologyData< shards::Quadrilateral<> >());  // create relevant subcell (side) topology
    shards::CellTopology line(shards::getCellTopologyData< shards::Line<> >());  // create relevant subcell (side) topology
    int cellDim = cell.getDimension();
    int sideDim = side.getDimension();
    unsigned numSides = 6;
 
    // Define array containing points at which the solution is evaluated, on the reference tet.
    int numIntervals = 10;
    int numInterpPoints = (numIntervals + 1)*(numIntervals + 1)*(numIntervals + 1);
    FieldContainer<double> interp_points_ref(numInterpPoints, 3);
    int counter = 0;
    for (int k=0; k<=numIntervals; k++) {
      for (int j=0; j<=numIntervals; j++) {
        for (int i=0; i<=numIntervals; i++) {
          interp_points_ref(counter,0) = i*(1.0/numIntervals)-1.0;
          interp_points_ref(counter,1) = j*(1.0/numIntervals)-1.0;
          interp_points_ref(counter,2) = k*(1.0/numIntervals)-1.0;
          counter++;
        }
      }
    }
 
    /* Parent cell definition. */
    FieldContainer<double> cell_nodes[2];
    cell_nodes[0].resize(1, 8, cellDim);
    cell_nodes[1].resize(1, 8, cellDim);
 
    // Generic parallelepiped.
    cell_nodes[0](0, 0, 0) = -5.0;
    cell_nodes[0](0, 0, 1) = -1.0;
    cell_nodes[0](0, 0, 2) = 0.0;
    cell_nodes[0](0, 1, 0) = 4.0;
    cell_nodes[0](0, 1, 1) = 1.0;
    cell_nodes[0](0, 1, 2) = 1.0;
    cell_nodes[0](0, 2, 0) = 8.0;
    cell_nodes[0](0, 2, 1) = 3.0;
    cell_nodes[0](0, 2, 2) = 1.0;
    cell_nodes[0](0, 3, 0) = -1.0;
    cell_nodes[0](0, 3, 1) = 1.0;
    cell_nodes[0](0, 3, 2) = 0.0;
    cell_nodes[0](0, 4, 0) = 5.0;
    cell_nodes[0](0, 4, 1) = 9.0;
    cell_nodes[0](0, 4, 2) = 1.0;
    cell_nodes[0](0, 5, 0) = 14.0;
    cell_nodes[0](0, 5, 1) = 11.0;
    cell_nodes[0](0, 5, 2) = 2.0;
    cell_nodes[0](0, 6, 0) = 18.0;
    cell_nodes[0](0, 6, 1) = 13.0;
    cell_nodes[0](0, 6, 2) = 2.0;
    cell_nodes[0](0, 7, 0) = 9.0;
    cell_nodes[0](0, 7, 1) = 11.0;
    cell_nodes[0](0, 7, 2) = 1.0;
    // Reference hex.
    cell_nodes[1](0, 0, 0) = -1.0;
    cell_nodes[1](0, 0, 1) = -1.0;
    cell_nodes[1](0, 0, 2) = -1.0;
    cell_nodes[1](0, 1, 0) = 1.0;
    cell_nodes[1](0, 1, 1) = -1.0;
    cell_nodes[1](0, 1, 2) = -1.0;
    cell_nodes[1](0, 2, 0) = 1.0;
    cell_nodes[1](0, 2, 1) = 1.0;
    cell_nodes[1](0, 2, 2) = -1.0;
    cell_nodes[1](0, 3, 0) = -1.0;
    cell_nodes[1](0, 3, 1) = 1.0;
    cell_nodes[1](0, 3, 2) = -1.0;
    cell_nodes[1](0, 4, 0) = -1.0;
    cell_nodes[1](0, 4, 1) = -1.0;
    cell_nodes[1](0, 4, 2) = 1.0;
    cell_nodes[1](0, 5, 0) = 1.0;
    cell_nodes[1](0, 5, 1) = -1.0;
    cell_nodes[1](0, 5, 2) = 1.0;
    cell_nodes[1](0, 6, 0) = 1.0;
    cell_nodes[1](0, 6, 1) = 1.0;
    cell_nodes[1](0, 6, 2) = 1.0;
    cell_nodes[1](0, 7, 0) = -1.0;
    cell_nodes[1](0, 7, 1) = 1.0;
    cell_nodes[1](0, 7, 2) = 1.0;
 
    std::stringstream mystream[2];
    mystream[0].str("\n>> Now testing basis on a generic parallelepiped ...\n");
    mystream[1].str("\n>> Now testing basis on the reference hex ...\n");
 
 
    for (int pcell = 0; pcell < 2; pcell++) {
      *outStream << mystream[pcell].str();
      FieldContainer<double> interp_points(1, numInterpPoints, cellDim);
      CellTools<double>::mapToPhysicalFrame(interp_points, interp_points_ref, cell_nodes[pcell], cell);
      interp_points.resize(numInterpPoints, cellDim);
 
      for (int x_order=0; x_order <= max_order; x_order++) {
        int max_y_order = max_order;
        if (pcell == 0) {
          max_y_order -= x_order;
        }
        for (int y_order=0; y_order <= max_y_order; y_order++) {
          int max_z_order = max_order;
          if (pcell == 0) {
            max_z_order -= x_order;
            max_z_order -= y_order;
          }
          for (int z_order=0; z_order <= max_z_order; z_order++) {
 
            // evaluate exact solution
            FieldContainer<double> exact_solution(1, numInterpPoints);
            u_exact(exact_solution, interp_points, x_order, y_order, z_order);
 
            int basis_order = max_order;
 
            // set test tolerance;
            double zero = basis_order*basis_order*basis_order*100*INTREPID_TOL;
 
            //create basis
            FieldContainer<double> pts(PointTools::getLatticeSize(line,basis_order),1);
            PointTools::getLattice<double,FieldContainer<double> >(pts,line,basis_order);
            
            Teuchos::RCP<Basis<double,FieldContainer<double> > > basis =
              Teuchos::rcp(new Basis_HGRAD_HEX_Cn_FEM<double,FieldContainer<double> >( basis_order, POINTTYPE_SPECTRAL ) );
            int numFields = basis->getCardinality();
 
            // create cubatures
            Teuchos::RCP<Cubature<double> > cellCub = cubFactory.create(cell, 2*basis_order);
            Teuchos::RCP<Cubature<double> > sideCub = cubFactory.create(side, 2*basis_order);
            int numCubPointsCell = cellCub->getNumPoints();
            int numCubPointsSide = sideCub->getNumPoints();
 
            /* Computational arrays. */
            /* Section 1: Related to parent cell integration. */
            FieldContainer<double> cub_points_cell(numCubPointsCell, cellDim);
            FieldContainer<double> cub_points_cell_physical(1, numCubPointsCell, cellDim);
            FieldContainer<double> cub_weights_cell(numCubPointsCell);
            FieldContainer<double> jacobian_cell(1, numCubPointsCell, cellDim, cellDim);
            FieldContainer<double> jacobian_inv_cell(1, numCubPointsCell, cellDim, cellDim);
            FieldContainer<double> jacobian_det_cell(1, numCubPointsCell);
            FieldContainer<double> weighted_measure_cell(1, numCubPointsCell);
 
            FieldContainer<double> value_of_basis_at_cub_points_cell(numFields, numCubPointsCell);
            FieldContainer<double> transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
            FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
            FieldContainer<double> grad_of_basis_at_cub_points_cell(numFields, numCubPointsCell, cellDim);
            FieldContainer<double> transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
            FieldContainer<double> weighted_transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
            FieldContainer<double> fe_matrix(1, numFields, numFields);
 
            FieldContainer<double> rhs_at_cub_points_cell_physical(1, numCubPointsCell);
            FieldContainer<double> rhs_and_soln_vector(1, numFields);
 
            /* Section 2: Related to subcell (side) integration. */
            FieldContainer<double> cub_points_side(numCubPointsSide, sideDim);
            FieldContainer<double> cub_weights_side(numCubPointsSide);
            FieldContainer<double> cub_points_side_refcell(numCubPointsSide, cellDim);
            FieldContainer<double> cub_points_side_physical(1, numCubPointsSide, cellDim);
            FieldContainer<double> jacobian_side_refcell(1, numCubPointsSide, cellDim, cellDim);
            FieldContainer<double> jacobian_det_side_refcell(1, numCubPointsSide);
            FieldContainer<double> weighted_measure_side_refcell(1, numCubPointsSide);
 
            FieldContainer<double> value_of_basis_at_cub_points_side_refcell(numFields, numCubPointsSide);
            FieldContainer<double> transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
            FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
            FieldContainer<double> neumann_data_at_cub_points_side_physical(1, numCubPointsSide);
            FieldContainer<double> neumann_fields_per_side(1, numFields);
 
            /* Section 3: Related to global interpolant. */
            FieldContainer<double> value_of_basis_at_interp_points_ref(numFields, numInterpPoints);
            FieldContainer<double> transformed_value_of_basis_at_interp_points_ref(1, numFields, numInterpPoints);
            FieldContainer<double> interpolant(1, numInterpPoints);
 
            FieldContainer<int> ipiv(numFields);
 
 
 
            /******************* START COMPUTATION ***********************/
 
            // get cubature points and weights
            cellCub->getCubature(cub_points_cell, cub_weights_cell);
 
            // compute geometric cell information
            CellTools<double>::setJacobian(jacobian_cell, cub_points_cell, cell_nodes[pcell], cell);
            CellTools<double>::setJacobianInv(jacobian_inv_cell, jacobian_cell);
            CellTools<double>::setJacobianDet(jacobian_det_cell, jacobian_cell);
 
            // compute weighted measure
            FunctionSpaceTools::computeCellMeasure<double>(weighted_measure_cell, jacobian_det_cell, cub_weights_cell);
 
            // Computing mass matrices:
            // tabulate values of basis functions at (reference) cubature points
            basis->getValues(value_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_VALUE);
 
            // transform values of basis functions 
            FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_cell,
                                                            value_of_basis_at_cub_points_cell);
 
            // multiply with weighted measure
            FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_cell,
                                                        weighted_measure_cell,
                                                        transformed_value_of_basis_at_cub_points_cell);
 
            // compute mass matrices
            FunctionSpaceTools::integrate<double>(fe_matrix,
                                                  transformed_value_of_basis_at_cub_points_cell,
                                                  weighted_transformed_value_of_basis_at_cub_points_cell,
                                                  COMP_BLAS);
 
            // Computing stiffness matrices:
            // tabulate gradients of basis functions at (reference) cubature points
            basis->getValues(grad_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_GRAD);
 
            // transform gradients of basis functions 
            FunctionSpaceTools::HGRADtransformGRAD<double>(transformed_grad_of_basis_at_cub_points_cell,
                                                           jacobian_inv_cell,
                                                           grad_of_basis_at_cub_points_cell);
 
            // multiply with weighted measure
            FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_grad_of_basis_at_cub_points_cell,
                                                        weighted_measure_cell,
                                                        transformed_grad_of_basis_at_cub_points_cell);
 
            // compute stiffness matrices and sum into fe_matrix
            FunctionSpaceTools::integrate<double>(fe_matrix,
                                                  transformed_grad_of_basis_at_cub_points_cell,
                                                  weighted_transformed_grad_of_basis_at_cub_points_cell,
                                                  COMP_BLAS,
                                                  true);
 
            // Computing RHS contributions:
            // map cell (reference) cubature points to physical space
            CellTools<double>::mapToPhysicalFrame(cub_points_cell_physical, cub_points_cell, cell_nodes[pcell], cell);
 
            // evaluate rhs function
            rhsFunc(rhs_at_cub_points_cell_physical, cub_points_cell_physical, x_order, y_order, z_order);
 
            // compute rhs
            FunctionSpaceTools::integrate<double>(rhs_and_soln_vector,
                                                  rhs_at_cub_points_cell_physical,
                                                  weighted_transformed_value_of_basis_at_cub_points_cell,
                                                  COMP_BLAS);
 
            // compute neumann b.c. contributions and adjust rhs
            sideCub->getCubature(cub_points_side, cub_weights_side);
            for (unsigned i=0; i<numSides; i++) {
              // compute geometric cell information
              CellTools<double>::mapToReferenceSubcell(cub_points_side_refcell, cub_points_side, sideDim, (int)i, cell);
              CellTools<double>::setJacobian(jacobian_side_refcell, cub_points_side_refcell, cell_nodes[pcell], cell);
              CellTools<double>::setJacobianDet(jacobian_det_side_refcell, jacobian_side_refcell);
 
              // compute weighted face measure
              FunctionSpaceTools::computeFaceMeasure<double>(weighted_measure_side_refcell,
                                                             jacobian_side_refcell,
                                                             cub_weights_side,
                                                             i,
                                                             cell);
 
              // tabulate values of basis functions at side cubature points, in the reference parent cell domain
              basis->getValues(value_of_basis_at_cub_points_side_refcell, cub_points_side_refcell, OPERATOR_VALUE);
              // transform 
              FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_side_refcell,
                                                              value_of_basis_at_cub_points_side_refcell);
 
              // multiply with weighted measure
              FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_side_refcell,
                                                          weighted_measure_side_refcell,
                                                          transformed_value_of_basis_at_cub_points_side_refcell);
 
              // compute Neumann data
              // map side cubature points in reference parent cell domain to physical space
              CellTools<double>::mapToPhysicalFrame(cub_points_side_physical, cub_points_side_refcell, cell_nodes[pcell], cell);
              // now compute data
              neumann(neumann_data_at_cub_points_side_physical, cub_points_side_physical, jacobian_side_refcell,
                      cell, (int)i, x_order, y_order, z_order);
 
              FunctionSpaceTools::integrate<double>(neumann_fields_per_side,
                                                    neumann_data_at_cub_points_side_physical,
                                                    weighted_transformed_value_of_basis_at_cub_points_side_refcell,
                                                    COMP_BLAS);
 
              // adjust RHS
              RealSpaceTools<double>::add(rhs_and_soln_vector, neumann_fields_per_side);;
            }
 
            // Solution of linear system:
            int info = 0;
            Teuchos::LAPACK<int, double> solver;
            solver.GESV(numFields, 1, &fe_matrix[0], numFields, &ipiv(0), &rhs_and_soln_vector[0], numFields, &info);
 
            // Building interpolant:
            // evaluate basis at interpolation points
            basis->getValues(value_of_basis_at_interp_points_ref, interp_points_ref, OPERATOR_VALUE);
            // transform values of basis functions 
            FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_interp_points_ref,
                                                            value_of_basis_at_interp_points_ref);
            FunctionSpaceTools::evaluate<double>(interpolant, rhs_and_soln_vector, transformed_value_of_basis_at_interp_points_ref);
 
            /******************* END COMPUTATION ***********************/
        
            RealSpaceTools<double>::subtract(interpolant, exact_solution);
 
            *outStream << "\nRelative norm-2 error between exact solution polynomial of order ("
                       << x_order << ", " << y_order << ", " << z_order
                       << ") and finite element interpolant of order " << basis_order << ": "
                       << RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
                          RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) << "\n";
 
            if (RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
                RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) > zero) {
              *outStream << "\n\nPatch test failed for solution polynomial order ("
                         << x_order << ", " << y_order << ", " << z_order << ") and basis order " << basis_order << "\n\n";
              errorFlag++;
            }
          } // end for z_order
        } // end for y_order
      } // end for x_order
    } // end for pcell
 
  }
  // Catch unexpected errors
  catch (std::logic_error err) {
    *outStream << err.what() << "\n\n";
    errorFlag = -1000;
  };
 
  if (errorFlag != 0)
    std::cout << "End Result: TEST FAILED\n";
  else
    std::cout << "End Result: TEST PASSED\n";
 
  // reset format state of std::cout
  std::cout.copyfmt(oldFormatState);
 
  return errorFlag;
}