Intrepid
/usr/src/RPM/BUILD/trilinos10-10.6.4/packages/intrepid/test/Discretization/Basis/HGRAD_TET_C1_FEM/test_02.cpp
00001 // @HEADER
00002 // ************************************************************************
00003 //
00004 //                           Intrepid Package
00005 //                 Copyright (2007) Sandia Corporation
00006 //
00007 // Under terms of Contract DE-AC04-94AL85000, there is a non-exclusive
00008 // license for use of this work by or on behalf of the U.S. Government.
00009 //
00010 // This library is free software; you can redistribute it and/or modify
00011 // it under the terms of the GNU Lesser General Public License as
00012 // published by the Free Software Foundation; either version 2.1 of the
00013 // License, or (at your option) any later version.
00014 //
00015 // This library is distributed in the hope that it will be useful, but
00016 // WITHOUT ANY WARRANTY; without even the implied warranty of
00017 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
00018 // Lesser General Public License for more details.
00019 //
00020 // You should have received a copy of the GNU Lesser General Public
00021 // License along with this library; if not, write to the Free Software
00022 // Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
00023 // USA
00024 // Questions? Contact Pavel Bochev  (pbboche@sandia.gov), 
00025 //                    Denis Ridzal  (dridzal@sandia.gov),
00026 //                    Kara Peterson (kjpeter@sandia.gov).
00027 //
00028 // ************************************************************************
00029 // @HEADER
00030 
00036 #include "Intrepid_FieldContainer.hpp"
00037 #include "Intrepid_HGRAD_TET_C1_FEM.hpp"
00038 #include "Intrepid_DefaultCubatureFactory.hpp"
00039 #include "Intrepid_RealSpaceTools.hpp"
00040 #include "Intrepid_ArrayTools.hpp"
00041 #include "Intrepid_FunctionSpaceTools.hpp"
00042 #include "Intrepid_CellTools.hpp"
00043 #include "Teuchos_oblackholestream.hpp"
00044 #include "Teuchos_RCP.hpp"
00045 #include "Teuchos_GlobalMPISession.hpp"
00046 #include "Teuchos_SerialDenseMatrix.hpp"
00047 #include "Teuchos_SerialDenseVector.hpp"
00048 #include "Teuchos_LAPACK.hpp"
00049 
00050 using namespace std;
00051 using namespace Intrepid;
00052 
00053 void rhsFunc(FieldContainer<double> &, const FieldContainer<double> &, int, int, int);
00054 void neumann(FieldContainer<double>       & ,
00055              const FieldContainer<double> & ,
00056              const FieldContainer<double> & ,
00057              const shards::CellTopology   & ,
00058              int, int, int, int);
00059 void u_exact(FieldContainer<double> &, const FieldContainer<double> &, int, int, int);
00060 
00062 void rhsFunc(FieldContainer<double> & result,
00063              const FieldContainer<double> & points,
00064              int xd,
00065              int yd,
00066              int zd) {
00067 
00068   int x = 0, y = 1, z = 2;
00069 
00070   // second x-derivatives of u
00071   if (xd > 1) {
00072     for (int cell=0; cell<result.dimension(0); cell++) {
00073       for (int pt=0; pt<result.dimension(1); pt++) {
00074         result(cell,pt) = - xd*(xd-1)*std::pow(points(cell,pt,x), xd-2) *
00075                             std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
00076       }
00077     }
00078   }
00079 
00080   // second y-derivatives of u
00081   if (yd > 1) {
00082     for (int cell=0; cell<result.dimension(0); cell++) {
00083       for (int pt=0; pt<result.dimension(1); pt++) {
00084         result(cell,pt) -=  yd*(yd-1)*std::pow(points(cell,pt,y), yd-2) *
00085                             std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,z), zd);
00086       }
00087     }
00088   }
00089 
00090   // second z-derivatives of u
00091   if (zd > 1) {
00092     for (int cell=0; cell<result.dimension(0); cell++) {
00093       for (int pt=0; pt<result.dimension(1); pt++) {
00094         result(cell,pt) -=  zd*(zd-1)*std::pow(points(cell,pt,z), zd-2) *
00095                             std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd);
00096       }
00097     }
00098   }
00099 
00100   // add u
00101   for (int cell=0; cell<result.dimension(0); cell++) {
00102     for (int pt=0; pt<result.dimension(1); pt++) {
00103       result(cell,pt) +=  std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
00104     }
00105   }
00106 
00107 }
00108 
00109 
00111 void neumann(FieldContainer<double>       & result,
00112              const FieldContainer<double> & points,
00113              const FieldContainer<double> & jacs,
00114              const shards::CellTopology   & parentCell,
00115              int sideOrdinal, int xd, int yd, int zd) {
00116 
00117   int x = 0, y = 1, z = 2;
00118 
00119   int numCells  = result.dimension(0);
00120   int numPoints = result.dimension(1);
00121 
00122   FieldContainer<double> grad_u(numCells, numPoints, 3);
00123   FieldContainer<double> side_normals(numCells, numPoints, 3);
00124   FieldContainer<double> normal_lengths(numCells, numPoints);
00125 
00126   // first x-derivatives of u
00127   if (xd > 0) {
00128     for (int cell=0; cell<numCells; cell++) {
00129       for (int pt=0; pt<numPoints; pt++) {
00130         grad_u(cell,pt,x) = xd*std::pow(points(cell,pt,x), xd-1) *
00131                             std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
00132       }
00133     }
00134   }
00135 
00136   // first y-derivatives of u
00137   if (yd > 0) {
00138     for (int cell=0; cell<numCells; cell++) {
00139       for (int pt=0; pt<numPoints; pt++) {
00140         grad_u(cell,pt,y) = yd*std::pow(points(cell,pt,y), yd-1) *
00141                             std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,z), zd);
00142       }
00143     }
00144   }
00145 
00146   // first z-derivatives of u
00147   if (zd > 0) {
00148     for (int cell=0; cell<numCells; cell++) {
00149       for (int pt=0; pt<numPoints; pt++) {
00150         grad_u(cell,pt,z) = zd*std::pow(points(cell,pt,z), zd-1) *
00151                             std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd);
00152       }
00153     }
00154   }
00155   
00156   CellTools<double>::getPhysicalSideNormals(side_normals, jacs, sideOrdinal, parentCell);
00157 
00158   // scale normals
00159   RealSpaceTools<double>::vectorNorm(normal_lengths, side_normals, NORM_TWO);
00160   FunctionSpaceTools::scalarMultiplyDataData<double>(side_normals, normal_lengths, side_normals, true); 
00161 
00162   FunctionSpaceTools::dotMultiplyDataData<double>(result, grad_u, side_normals);
00163 
00164 }
00165 
00167 void u_exact(FieldContainer<double> & result, const FieldContainer<double> & points, int xd, int yd, int zd) {
00168   int x = 0, y = 1, z = 2;
00169   for (int cell=0; cell<result.dimension(0); cell++) {
00170     for (int pt=0; pt<result.dimension(1); pt++) {
00171       result(cell,pt) = std::pow(points(pt,x), xd)*std::pow(points(pt,y), yd)*std::pow(points(pt,z), zd);
00172     }
00173   }
00174 }
00175 
00176 
00177 
00178 
00179 int main(int argc, char *argv[]) {
00180 
00181   Teuchos::GlobalMPISession mpiSession(&argc, &argv);
00182 
00183   // This little trick lets us print to std::cout only if
00184   // a (dummy) command-line argument is provided.
00185   int iprint     = argc - 1;
00186   Teuchos::RCP<std::ostream> outStream;
00187   Teuchos::oblackholestream bhs; // outputs nothing
00188   if (iprint > 0)
00189     outStream = Teuchos::rcp(&std::cout, false);
00190   else
00191     outStream = Teuchos::rcp(&bhs, false);
00192 
00193   // Save the format state of the original std::cout.
00194   Teuchos::oblackholestream oldFormatState;
00195   oldFormatState.copyfmt(std::cout);
00196 
00197   *outStream \
00198     << "===============================================================================\n" \
00199     << "|                                                                             |\n" \
00200     << "|                    Unit Test (Basis_HGRAD_TET_C1_FEM)                       |\n" \
00201     << "|                                                                             |\n" \
00202     << "|     1) Patch test involving mass and stiffness matrices,                    |\n" \
00203     << "|        for the Neumann problem on a tetrahedral patch                       |\n" \
00204     << "|        Omega with boundary Gamma.                                           |\n" \
00205     << "|                                                                             |\n" \
00206     << "|        - div (grad u) + u = f  in Omega,  (grad u) . n = g  on Gamma        |\n" \
00207     << "|                                                                             |\n" \
00208     << "|  Questions? Contact  Pavel Bochev  (pbboche@sandia.gov),                    |\n" \
00209     << "|                      Denis Ridzal  (dridzal@sandia.gov),                    |\n" \
00210     << "|                      Kara Peterson (kjpeter@sandia.gov).                    |\n" \
00211     << "|                                                                             |\n" \
00212     << "|  Intrepid's website: http://trilinos.sandia.gov/packages/intrepid           |\n" \
00213     << "|  Trilinos website:   http://trilinos.sandia.gov                             |\n" \
00214     << "|                                                                             |\n" \
00215     << "===============================================================================\n"\
00216     << "| TEST 1: Patch test                                                          |\n"\
00217     << "===============================================================================\n";
00218 
00219   
00220   int errorFlag = 0;
00221 
00222   outStream -> precision(16);
00223 
00224 
00225   try {
00226 
00227     int max_order = 1;                                                                  // max total order of polynomial solution
00228     DefaultCubatureFactory<double>  cubFactory;                                         // create factory
00229     shards::CellTopology cell(shards::getCellTopologyData< shards::Tetrahedron<> >());  // create parent cell topology
00230     shards::CellTopology side(shards::getCellTopologyData< shards::Triangle<> >());     // create relevant subcell (side) topology
00231     int cellDim = cell.getDimension();
00232     int sideDim = side.getDimension();
00233 
00234     // Define array containing points at which the solution is evaluated, on the reference tet.
00235     int numIntervals = 10;
00236     int numInterpPoints = ((numIntervals + 1)*(numIntervals + 2)*(numIntervals + 3))/6;
00237     FieldContainer<double> interp_points_ref(numInterpPoints, 3);
00238     int counter = 0;
00239     for (int k=0; k<=numIntervals; k++) {
00240       for (int j=0; j<=numIntervals; j++) {
00241         for (int i=0; i<=numIntervals; i++) {
00242           if (i+j+k <= numIntervals) {
00243             interp_points_ref(counter,0) = i*(1.0/numIntervals);
00244             interp_points_ref(counter,1) = j*(1.0/numIntervals);
00245             interp_points_ref(counter,2) = k*(1.0/numIntervals);
00246             counter++;
00247           }
00248         }
00249       }
00250     }
00251 
00252     /* Definition of parent cell. */
00253     FieldContainer<double> cell_nodes(1, 4, cellDim);
00254     // funky tet
00255     cell_nodes(0, 0, 0) = -1.0;
00256     cell_nodes(0, 0, 1) = -2.0;
00257     cell_nodes(0, 0, 2) = 0.0;
00258     cell_nodes(0, 1, 0) = 6.0;
00259     cell_nodes(0, 1, 1) = 2.0;
00260     cell_nodes(0, 1, 2) = 0.0;
00261     cell_nodes(0, 2, 0) = -5.0;
00262     cell_nodes(0, 2, 1) = 1.0;
00263     cell_nodes(0, 2, 2) = 0.0;
00264     cell_nodes(0, 3, 0) = -4.0;
00265     cell_nodes(0, 3, 1) = -1.0;
00266     cell_nodes(0, 3, 2) = 3.0;
00267     // perturbed reference tet
00268     /*cell_nodes(0, 0, 0) = 0.1;
00269     cell_nodes(0, 0, 1) = -0.1;
00270     cell_nodes(0, 0, 2) = 0.2;
00271     cell_nodes(0, 1, 0) = 1.2;
00272     cell_nodes(0, 1, 1) = -0.1;
00273     cell_nodes(0, 1, 2) = 0.05;
00274     cell_nodes(0, 2, 0) = 0.0;
00275     cell_nodes(0, 2, 1) = 0.9;
00276     cell_nodes(0, 2, 2) = 0.1;
00277     cell_nodes(0, 3, 0) = 0.1;
00278     cell_nodes(0, 3, 1) = -0.1;
00279     cell_nodes(0, 3, 2) = 1.1;*/
00280     // reference tet
00281     /*cell_nodes(0, 0, 0) = 0.0;
00282     cell_nodes(0, 0, 1) = 0.0;
00283     cell_nodes(0, 0, 2) = 0.0;
00284     cell_nodes(0, 1, 0) = 1.0;
00285     cell_nodes(0, 1, 1) = 0.0;
00286     cell_nodes(0, 1, 2) = 0.0;
00287     cell_nodes(0, 2, 0) = 0.0;
00288     cell_nodes(0, 2, 1) = 1.0;
00289     cell_nodes(0, 2, 2) = 0.0;
00290     cell_nodes(0, 3, 0) = 0.0;
00291     cell_nodes(0, 3, 1) = 0.0;
00292     cell_nodes(0, 3, 2) = 1.0;*/
00293 
00294     FieldContainer<double> interp_points(1, numInterpPoints, cellDim);
00295     CellTools<double>::mapToPhysicalFrame(interp_points, interp_points_ref, cell_nodes, cell);
00296     interp_points.resize(numInterpPoints, cellDim);
00297 
00298     for (int x_order=0; x_order <= max_order; x_order++) {
00299       for (int y_order=0; y_order <= max_order-x_order; y_order++) {
00300         for (int z_order=0; z_order <= max_order-x_order-y_order; z_order++) {
00301 
00302           // evaluate exact solution
00303           FieldContainer<double> exact_solution(1, numInterpPoints);
00304           u_exact(exact_solution, interp_points, x_order, y_order, z_order);
00305 
00306           int basis_order = 1;
00307 
00308           // set test tolerance;
00309           double zero = basis_order*basis_order*basis_order*100*INTREPID_TOL;
00310 
00311           //create basis
00312           Teuchos::RCP<Basis<double,FieldContainer<double> > > basis =
00313             Teuchos::rcp(new Basis_HGRAD_TET_C1_FEM<double,FieldContainer<double> >() );
00314           int numFields = basis->getCardinality();
00315 
00316           // create cubatures
00317           Teuchos::RCP<Cubature<double> > cellCub = cubFactory.create(cell, 2*basis_order);
00318           Teuchos::RCP<Cubature<double> > sideCub = cubFactory.create(side, 2*basis_order);
00319           int numCubPointsCell = cellCub->getNumPoints();
00320           int numCubPointsSide = sideCub->getNumPoints();
00321 
00322           /* Computational arrays. */
00323           /* Section 1: Related to parent cell integration. */
00324           FieldContainer<double> cub_points_cell(numCubPointsCell, cellDim);
00325           FieldContainer<double> cub_points_cell_physical(1, numCubPointsCell, cellDim);
00326           FieldContainer<double> cub_weights_cell(numCubPointsCell);
00327           FieldContainer<double> jacobian_cell(1, numCubPointsCell, cellDim, cellDim);
00328           FieldContainer<double> jacobian_inv_cell(1, numCubPointsCell, cellDim, cellDim);
00329           FieldContainer<double> jacobian_det_cell(1, numCubPointsCell);
00330           FieldContainer<double> weighted_measure_cell(1, numCubPointsCell);
00331 
00332           FieldContainer<double> value_of_basis_at_cub_points_cell(numFields, numCubPointsCell);
00333           FieldContainer<double> transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
00334           FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
00335           FieldContainer<double> grad_of_basis_at_cub_points_cell(numFields, numCubPointsCell, cellDim);
00336           FieldContainer<double> transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
00337           FieldContainer<double> weighted_transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
00338           FieldContainer<double> fe_matrix(1, numFields, numFields);
00339 
00340           FieldContainer<double> rhs_at_cub_points_cell_physical(1, numCubPointsCell);
00341           FieldContainer<double> rhs_and_soln_vector(1, numFields);
00342 
00343           /* Section 2: Related to subcell (side) integration. */
00344           unsigned numSides = 4;
00345           FieldContainer<double> cub_points_side(numCubPointsSide, sideDim);
00346           FieldContainer<double> cub_weights_side(numCubPointsSide);
00347           FieldContainer<double> cub_points_side_refcell(numCubPointsSide, cellDim);
00348           FieldContainer<double> cub_points_side_physical(1, numCubPointsSide, cellDim);
00349           FieldContainer<double> jacobian_side_refcell(1, numCubPointsSide, cellDim, cellDim);
00350           FieldContainer<double> jacobian_det_side_refcell(1, numCubPointsSide);
00351           FieldContainer<double> weighted_measure_side_refcell(1, numCubPointsSide);
00352 
00353           FieldContainer<double> value_of_basis_at_cub_points_side_refcell(numFields, numCubPointsSide);
00354           FieldContainer<double> transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
00355           FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
00356           FieldContainer<double> neumann_data_at_cub_points_side_physical(1, numCubPointsSide);
00357           FieldContainer<double> neumann_fields_per_side(1, numFields);
00358 
00359           /* Section 3: Related to global interpolant. */
00360           FieldContainer<double> value_of_basis_at_interp_points_ref(numFields, numInterpPoints);
00361           FieldContainer<double> transformed_value_of_basis_at_interp_points_ref(1, numFields, numInterpPoints);
00362           FieldContainer<double> interpolant(1, numInterpPoints);
00363 
00364           FieldContainer<int> ipiv(numFields);
00365 
00366 
00367 
00368           /******************* START COMPUTATION ***********************/
00369 
00370           // get cubature points and weights
00371           cellCub->getCubature(cub_points_cell, cub_weights_cell);
00372 
00373           // compute geometric cell information
00374           CellTools<double>::setJacobian(jacobian_cell, cub_points_cell, cell_nodes, cell);
00375           CellTools<double>::setJacobianInv(jacobian_inv_cell, jacobian_cell);
00376           CellTools<double>::setJacobianDet(jacobian_det_cell, jacobian_cell);
00377 
00378           // compute weighted measure
00379           FunctionSpaceTools::computeCellMeasure<double>(weighted_measure_cell, jacobian_det_cell, cub_weights_cell);
00380 
00382           // Computing mass matrices:
00383           // tabulate values of basis functions at (reference) cubature points
00384           basis->getValues(value_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_VALUE);
00385 
00386           // transform values of basis functions 
00387           FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_cell,
00388                                                           value_of_basis_at_cub_points_cell);
00389 
00390           // multiply with weighted measure
00391           FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_cell,
00392                                                       weighted_measure_cell,
00393                                                       transformed_value_of_basis_at_cub_points_cell);
00394 
00395           // compute mass matrices
00396           FunctionSpaceTools::integrate<double>(fe_matrix,
00397                                                 transformed_value_of_basis_at_cub_points_cell,
00398                                                 weighted_transformed_value_of_basis_at_cub_points_cell,
00399                                                 COMP_BLAS);
00401 
00403           // Computing stiffness matrices:
00404           // tabulate gradients of basis functions at (reference) cubature points
00405           basis->getValues(grad_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_GRAD);
00406 
00407           // transform gradients of basis functions 
00408           FunctionSpaceTools::HGRADtransformGRAD<double>(transformed_grad_of_basis_at_cub_points_cell,
00409                                                          jacobian_inv_cell,
00410                                                          grad_of_basis_at_cub_points_cell);
00411 
00412           // multiply with weighted measure
00413           FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_grad_of_basis_at_cub_points_cell,
00414                                                       weighted_measure_cell,
00415                                                       transformed_grad_of_basis_at_cub_points_cell);
00416 
00417           // compute stiffness matrices and sum into fe_matrix
00418           FunctionSpaceTools::integrate<double>(fe_matrix,
00419                                                 transformed_grad_of_basis_at_cub_points_cell,
00420                                                 weighted_transformed_grad_of_basis_at_cub_points_cell,
00421                                                 COMP_BLAS,
00422                                                 true);
00424 
00426           // Computing RHS contributions:
00427           // map cell (reference) cubature points to physical space
00428           CellTools<double>::mapToPhysicalFrame(cub_points_cell_physical, cub_points_cell, cell_nodes, cell);
00429 
00430           // evaluate rhs function
00431           rhsFunc(rhs_at_cub_points_cell_physical, cub_points_cell_physical, x_order, y_order, z_order);
00432 
00433           // compute rhs
00434           FunctionSpaceTools::integrate<double>(rhs_and_soln_vector,
00435                                                 rhs_at_cub_points_cell_physical,
00436                                                 weighted_transformed_value_of_basis_at_cub_points_cell,
00437                                                 COMP_BLAS);
00438 
00439           // compute neumann b.c. contributions and adjust rhs
00440           sideCub->getCubature(cub_points_side, cub_weights_side);
00441           for (unsigned i=0; i<numSides; i++) {
00442             // compute geometric cell information
00443             CellTools<double>::mapToReferenceSubcell(cub_points_side_refcell, cub_points_side, sideDim, (int)i, cell);
00444             CellTools<double>::setJacobian(jacobian_side_refcell, cub_points_side_refcell, cell_nodes, cell);
00445             CellTools<double>::setJacobianDet(jacobian_det_side_refcell, jacobian_side_refcell);
00446 
00447             // compute weighted face measure
00448             FunctionSpaceTools::computeFaceMeasure<double>(weighted_measure_side_refcell,
00449                                                            jacobian_side_refcell,
00450                                                            cub_weights_side,
00451                                                            i,
00452                                                            cell);
00453 
00454             // tabulate values of basis functions at side cubature points, in the reference parent cell domain
00455             basis->getValues(value_of_basis_at_cub_points_side_refcell, cub_points_side_refcell, OPERATOR_VALUE);
00456             // transform 
00457             FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_side_refcell,
00458                                                             value_of_basis_at_cub_points_side_refcell);
00459 
00460             // multiply with weighted measure
00461             FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_side_refcell,
00462                                                         weighted_measure_side_refcell,
00463                                                         transformed_value_of_basis_at_cub_points_side_refcell);
00464 
00465             // compute Neumann data
00466             // map side cubature points in reference parent cell domain to physical space
00467             CellTools<double>::mapToPhysicalFrame(cub_points_side_physical, cub_points_side_refcell, cell_nodes, cell);
00468             // now compute data
00469             neumann(neumann_data_at_cub_points_side_physical, cub_points_side_physical, jacobian_side_refcell,
00470                     cell, (int)i, x_order, y_order, z_order);
00471 
00472             FunctionSpaceTools::integrate<double>(neumann_fields_per_side,
00473                                                   neumann_data_at_cub_points_side_physical,
00474                                                   weighted_transformed_value_of_basis_at_cub_points_side_refcell,
00475                                                   COMP_BLAS);
00476 
00477             // adjust RHS
00478             RealSpaceTools<double>::add(rhs_and_soln_vector, neumann_fields_per_side);;
00479           }
00481 
00483           // Solution of linear system:
00484           int info = 0;
00485           Teuchos::LAPACK<int, double> solver;
00486           solver.GESV(numFields, 1, &fe_matrix[0], numFields, &ipiv(0), &rhs_and_soln_vector[0], numFields, &info);
00488 
00490           // Building interpolant:
00491           // evaluate basis at interpolation points
00492           basis->getValues(value_of_basis_at_interp_points_ref, interp_points_ref, OPERATOR_VALUE);
00493           // transform values of basis functions 
00494           FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_interp_points_ref,
00495                                                           value_of_basis_at_interp_points_ref);
00496           FunctionSpaceTools::evaluate<double>(interpolant, rhs_and_soln_vector, transformed_value_of_basis_at_interp_points_ref);
00498 
00499           /******************* END COMPUTATION ***********************/
00500       
00501           RealSpaceTools<double>::subtract(interpolant, exact_solution);
00502 
00503           *outStream << "\nRelative norm-2 error between exact solution polynomial of order ("
00504                      << x_order << ", " << y_order << ", " << z_order
00505                      << ") and finite element interpolant of order " << basis_order << ": "
00506                      << RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
00507                         RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) << "\n";
00508 
00509           if (RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
00510               RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) > zero) {
00511             *outStream << "\n\nPatch test failed for solution polynomial order ("
00512                        << x_order << ", " << y_order << ", " << z_order << ") and basis order " << basis_order << "\n\n";
00513             errorFlag++;
00514           }
00515         } // end for z_order
00516       } // end for y_order
00517     } // end for x_order
00518 
00519   }
00520   // Catch unexpected errors
00521   catch (std::logic_error err) {
00522     *outStream << err.what() << "\n\n";
00523     errorFlag = -1000;
00524   };
00525 
00526   if (errorFlag != 0)
00527     std::cout << "End Result: TEST FAILED\n";
00528   else
00529     std::cout << "End Result: TEST PASSED\n";
00530 
00531   // reset format state of std::cout
00532   std::cout.copyfmt(oldFormatState);
00533 
00534   return errorFlag;
00535 }