http://trilinos.sandia.gov/packages/docs/r10.0/packages/intrepid/test/Discretization/Basis/HGRAD_QUAD_C2_FEM/test_02.cpp

// @HEADER
// ************************************************************************
//
//                           Intrepid Package
//                 Copyright (2007) Sandia Corporation
//
// Under terms of Contract DE-AC04-94AL85000, there is a non-exclusive
// license for use of this work by or on behalf of the U.S. Government.
//
// This library is free software; you can redistribute it and/or modify
// it under the terms of the GNU Lesser General Public License as
// published by the Free Software Foundation; either version 2.1 of the
// License, or (at your option) any later version.
//
// This library is distributed in the hope that it will be useful, but
// WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
// USA
// Questions? Contact Pavel Bochev  (pbboche@sandia.gov), 
//                    Denis Ridzal  (dridzal@sandia.gov),
//                    Kara Peterson (kjpeter@sandia.gov).
//
// ************************************************************************
// @HEADER

#include "Intrepid_FieldContainer.hpp"
#include "Intrepid_HGRAD_QUAD_C2_FEM.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);
void neumann(FieldContainer<double>       & ,
             const FieldContainer<double> & ,
             const FieldContainer<double> & ,
             const shards::CellTopology   & ,
             int, int, int);
void u_exact(FieldContainer<double> &, const FieldContainer<double> &, int, int);

void rhsFunc(FieldContainer<double> & result,
             const FieldContainer<double> & points,
             int xd,
             int yd) {

  int x = 0, y = 1;

  // 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);
      }
    }
  }

  // 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);
      }
    }
  }

  // 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);
    }
  }

}


void neumann(FieldContainer<double>       & result,
             const FieldContainer<double> & points,
             const FieldContainer<double> & jacs,
             const shards::CellTopology   & parentCell,
             int sideOrdinal, int xd, int yd) {

  int x = 0, y = 1;

  int numCells  = result.dimension(0);
  int numPoints = result.dimension(1);

  FieldContainer<double> grad_u(numCells, numPoints, 2);
  FieldContainer<double> side_normals(numCells, numPoints, 2);
  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);
      }
    }
  }

  // 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);
      }
    }
  }
  
  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 x = 0, y = 1;
  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);
    }
  }
}




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_QUAD_C2_FEM)                      |\n" \
    << "|                                                                             |\n" \
    << "|     1) Patch test involving mass and stiffness matrices,                    |\n" \
    << "|        for the Neumann problem on a physical parallelogram                  |\n" \
    << "|        AND a reference quad Omega with boundary Gamma.                      |\n" \
    << "|                                                                             |\n" \
    << "|        - div (grad u) + u = f  in Omega,  (grad u) . n = g  on Gamma        |\n" \
    << "|                                                                             |\n" \
    << "|        For a generic parallelogram, the basis recovers a complete           |\n" \
    << "|        polynomial space of order 2. On a (scaled and/or translated)         |\n" \
    << "|        reference quad, the basis recovers a complete tensor product         |\n" \
    << "|        space of order 2 (i.e. incl. the x^2*y, x*y^2, x^2*y^2 terms).       |\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 = 2;                                                                    // max total order of polynomial solution
    DefaultCubatureFactory<double>  cubFactory;                                           // create cubature factory
    shards::CellTopology cell(shards::getCellTopologyData< shards::Quadrilateral<> >());  // create parent cell topology
    shards::CellTopology side(shards::getCellTopologyData< shards::Line<> >());           // create relevant subcell (side) topology
    int cellDim = cell.getDimension();
    int sideDim = side.getDimension();

    // Define array containing points at which the solution is evaluated, in reference cell.
    int numIntervals = 10;
    int numInterpPoints = (numIntervals + 1)*(numIntervals + 1);
    FieldContainer<double> interp_points_ref(numInterpPoints, 2);
    int counter = 0;
    for (int j=0; j<=numIntervals; j++) {
      for (int i=0; i<=numIntervals; i++) {
        interp_points_ref(counter,0) = i*(2.0/numIntervals)-1.0;
        interp_points_ref(counter,1) = j*(2.0/numIntervals)-1.0;
        counter++;
      }
    }

    /* Parent cell definition. */
    FieldContainer<double> cell_nodes[2];
    cell_nodes[0].resize(1, 4, cellDim);
    cell_nodes[1].resize(1, 4, cellDim);

    // Generic parallelogram.
    cell_nodes[0](0, 0, 0) = -5.0;
    cell_nodes[0](0, 0, 1) = -1.0;
    cell_nodes[0](0, 1, 0) = 4.0;
    cell_nodes[0](0, 1, 1) = 1.0;
    cell_nodes[0](0, 2, 0) = 8.0;
    cell_nodes[0](0, 2, 1) = 3.0;
    cell_nodes[0](0, 3, 0) = -1.0;
    cell_nodes[0](0, 3, 1) = 1.0;
    // Reference quad. 
    cell_nodes[1](0, 0, 0) = -1.0;
    cell_nodes[1](0, 0, 1) = -1.0;
    cell_nodes[1](0, 1, 0) = 1.0;
    cell_nodes[1](0, 1, 1) = -1.0;
    cell_nodes[1](0, 2, 0) = 1.0;
    cell_nodes[1](0, 2, 1) = 1.0;
    cell_nodes[1](0, 3, 0) = -1.0;
    cell_nodes[1](0, 3, 1) = 1.0;

    std::stringstream mystream[2];
    mystream[0].str("\n>> Now testing basis on a generic parallelogram ...\n");
    mystream[1].str("\n>> Now testing basis on the reference quad ...\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++) {

          // evaluate exact solution
          FieldContainer<double> exact_solution(1, numInterpPoints);
          u_exact(exact_solution, interp_points, x_order, y_order);

          int basis_order = 2;

          // set test tolerance
          double zero = basis_order*basis_order*100*INTREPID_TOL;

          //create basis
          Teuchos::RCP<Basis<double,FieldContainer<double> > > basis =
            Teuchos::rcp(new Basis_HGRAD_QUAD_C2_FEM<double,FieldContainer<double> >() );
          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. */
          unsigned numSides = 4;
          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(numFields, numInterpPoints);
          FieldContainer<double> transformed_value_of_basis_at_interp_points(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);

          // 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 edge measure
            FunctionSpaceTools::computeEdgeMeasure<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);

            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, interp_points_ref, OPERATOR_VALUE);
          // transform values of basis functions
          FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_interp_points,
                                                          value_of_basis_at_interp_points);
          FunctionSpaceTools::evaluate<double>(interpolant, rhs_and_soln_vector, transformed_value_of_basis_at_interp_points);

          /******************* END COMPUTATION ***********************/
      
          RealSpaceTools<double>::subtract(interpolant, exact_solution);

          *outStream << "\nRelative norm-2 error between exact solution polynomial of order ("
                     << x_order << ", " << y_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 << ") and basis order " << basis_order << "\n\n";
            errorFlag++;
          }
        } // 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;
}

---