bool LinearTermTests::testSums()
{
  bool success = true;

  LinearTermPtr sum = v1 + v2;

  if (sum->summands().size() != 2)
  {
    success = false;
    cout << "sum has the wrong number of summands\n";
    return success;
  }

  LinearSummand first_summand = sum->summands()[0];
  LinearSummand second_summand = sum->summands()[1];

  VarPtr first_var = first_summand.second;
  Camellia::EOperator first_op = first_var->op();

  VarPtr second_var = second_summand.second;
  Camellia::EOperator second_op = second_var->op();

  if (first_var->ID() != v1->ID())
  {
    success = false;
    cout << "first summand isn't v1.\n";
  }

  if (first_var->op() != OP_VALUE)
  {
    success = false;
    cout << "first op isn't VALUE.\n";
  }

  if (second_var->ID() != v2->ID())
  {
    success = false;
    cout << "second summand isn't v2 (is named " << second_var->name() << ").\n";
  }

  if (second_var->op() != OP_VALUE)
  {
    success = false;
    cout << "second op isn't VALUE.\n";
  }

  // check that sum reports having both varIDs
  if (sum->varIDs().find(v1->ID()) == sum->varIDs().end())
  {
    cout << "sum->varIDs() doesn't include v1.\n";
    success = false;
  }

  if (sum->varIDs().find(v2->ID()) == sum->varIDs().end())
  {
    cout << "sum->varIDs() doesn't include v2.\n";
    success = false;
  }

  if (sum->varIDs().size() != 2)
  {
    cout << "sum->varIDs() doesn't have the expected size (expected 2; is " << sum->varIDs().size() << ").\n";
    success = false;
  }

  // TODO: check that the sum is correct

  return success;
}
Beispiel #2
0
    virtual void localStiffnessMatrixAndRHS(FieldContainer<double> &localStiffness, FieldContainer<double> &rhsVector,
                                            IPPtr ip, BasisCachePtr ipBasisCache,
                                            RHSPtr rhs, BasisCachePtr basisCache) {
        double testMatrixAssemblyTime = 0, testMatrixInversionTime = 0, localStiffnessDeterminationFromTestsTime = 0;

#ifdef HAVE_MPI
        Epetra_MpiComm Comm(MPI_COMM_WORLD);
        //cout << "rank: " << rank << " of " << numProcs << endl;
#else
        Epetra_SerialComm Comm;
#endif

        Epetra_Time timer(Comm);

        // localStiffness should have dim. (numCells, numTrialFields, numTrialFields)
        MeshPtr mesh = basisCache->mesh();
        if (mesh.get() == NULL) {
            cout << "localStiffnessMatrix requires BasisCache to have mesh set.\n";
            TEUCHOS_TEST_FOR_EXCEPTION(true, std::invalid_argument, "localStiffnessMatrix requires BasisCache to have mesh set.");
        }
        const vector<GlobalIndexType>* cellIDs = &basisCache->cellIDs();
        int numCells = cellIDs->size();
        if (numCells != localStiffness.dimension(0)) {
            cout << "localStiffnessMatrix requires basisCache->cellIDs() to have the same # of cells as the first dimension of localStiffness\n";
            TEUCHOS_TEST_FOR_EXCEPTION(true, std::invalid_argument, "localStiffnessMatrix requires basisCache->cellIDs() to have the same # of cells as the first dimension of localStiffness");
        }

        ElementTypePtr elemType = mesh->getElementType((*cellIDs)[0]); // we assume all cells provided are of the same type
        DofOrderingPtr trialOrder = elemType->trialOrderPtr;
        DofOrderingPtr fieldOrder = mesh->getDofOrderingFactory().getFieldOrdering(trialOrder);
        DofOrderingPtr traceOrder = mesh->getDofOrderingFactory().getTraceOrdering(trialOrder);

        map<int, int> stiffnessIndexForTraceIndex;
        map<int, int> stiffnessIndexForFieldIndex;
        set<int> varIDs = trialOrder->getVarIDs();
        for (set<int>::iterator varIt = varIDs.begin(); varIt != varIDs.end(); varIt++) {
            int varID = *varIt;
            const vector<int>* sidesForVar = &trialOrder->getSidesForVarID(varID);
            bool isTrace = (sidesForVar->size() > 1);
            for (vector<int>::const_iterator sideIt = sidesForVar->begin(); sideIt != sidesForVar->end(); sideIt++) {
                int sideOrdinal = *sideIt;
                vector<int> dofIndices = trialOrder->getDofIndices(varID,sideOrdinal);
                if (isTrace) {
                    vector<int> traceDofIndices = traceOrder->getDofIndices(varID,sideOrdinal);
                    for (int i=0; i<traceDofIndices.size(); i++) {
                        stiffnessIndexForTraceIndex[traceDofIndices[i]] = dofIndices[i];
                    }
                } else {
                    vector<int> fieldDofIndices = fieldOrder->getDofIndices(varID);
                    for (int i=0; i<fieldDofIndices.size(); i++) {
                        stiffnessIndexForFieldIndex[fieldDofIndices[i]] = dofIndices[i];
                    }
                }
            }
        }

        int numTrialDofs = trialOrder->totalDofs();
        if ((numTrialDofs != localStiffness.dimension(1)) || (numTrialDofs != localStiffness.dimension(2))) {
            cout << "localStiffness should have dimensions (C,numTrialFields,numTrialFields).\n";
            TEUCHOS_TEST_FOR_EXCEPTION(true, std::invalid_argument, "localStiffness should have dimensions (C,numTrialFields,numTrialFields).");
        }

        map<int,int> traceTestMap, fieldTestMap;
        int numEquations = _virtualTerms.getFieldTestVars().size();
        for (int eqn=0; eqn<numEquations; eqn++) {
            VarPtr testVar = _virtualTerms.getFieldTestVars()[eqn];
            VarPtr traceVar = _virtualTerms.getAssociatedTrace(testVar);
            VarPtr fieldVar = _virtualTerms.getAssociatedField(testVar);
            traceTestMap[traceVar->ID()] = testVar->ID();
            fieldTestMap[fieldVar->ID()] = testVar->ID();
        }

        int maxDegreeField = fieldOrder->maxBasisDegree();
        int testDegreeInterior = maxDegreeField + _virtualTerms.getTestEnrichment();
        int testDegreeTrace = testDegreeInterior + 2;

        cout << "ERROR in Virtual: getRelabeledDofOrdering() is commented out in DofOrderingFactory.  Need to rewrite for the new caching scheme.\n";
        DofOrderingPtr testOrderInterior; // = mesh->getDofOrderingFactory().getRelabeledDofOrdering(fieldOrder, fieldTestMap);
        testOrderInterior = mesh->getDofOrderingFactory().setBasisDegree(testOrderInterior, testDegreeInterior, false);
        DofOrderingPtr testOrderTrace = mesh->getDofOrderingFactory().setBasisDegree(testOrderInterior, testDegreeTrace, true); // this has a bunch of extra dofs (interior guys)

        map<int, int> remappedTraceIndices; // go from the index that includes the interior dofs to one that doesn't
        set<int> testIDs = testOrderTrace->getVarIDs();
        int testTraceIndex = 0;
        for (set<int>::iterator testIDIt = testIDs.begin(); testIDIt != testIDs.end(); testIDIt++) {
            int testID = *testIDIt;
            BasisPtr basis = testOrderTrace->getBasis(testID);
            vector<int> interiorDofs = basis->dofOrdinalsForInterior();
            for (int basisOrdinal=0; basisOrdinal<basis->getCardinality(); basisOrdinal++) {
                if (std::find(interiorDofs.begin(),interiorDofs.end(),basisOrdinal) == interiorDofs.end()) {
                    int dofIndex = testOrderTrace->getDofIndex(testID, basisOrdinal);
                    remappedTraceIndices[dofIndex] = testTraceIndex;
                    testTraceIndex++;
                }
            }
        }

//    DofOrderingPtr testOrderTrace = mesh->getDofOrderingFactory().getRelabeledDofOrdering(traceOrder, traceTestMap);
//    testOrderTrace = mesh->getDofOrderingFactory().setBasisDegree(testOrderTrace, testDegreeTrace);

        int numTestInteriorDofs = testOrderInterior->totalDofs();
        int numTestTraceDofsIncludingInterior = testOrderTrace->totalDofs();
        int numTestTraceDofs = testTraceIndex;
        int numTestDofs = numTestTraceDofs + numTestInteriorDofs;

        timer.ResetStartTime();

        bool printTimings = true;

        if (printTimings) {
            cout << "numCells: " << numCells << endl;
            cout << "numTestDofs: " << numTestDofs << endl;
        }

        FieldContainer<double> rhsVectorTest(numCells,testOrderInterior->totalDofs()); // rhsVector is zero for the "trace" test dofs
        {
            // project the load f onto the space of interior test dofs.
            LinearTermPtr f = rhs->linearTerm();
            set<int> testIDs = f->varIDs();
            for (int eqn=0; eqn<numEquations; eqn++) {
                VarPtr v = _virtualTerms.getFieldTestVars()[eqn];

                if (testIDs.find(v->ID()) != testIDs.end()) {
                    BasisPtr testInteriorBasis = testOrderInterior->getBasis(v->ID());
                    FieldContainer<double> fValues(numCells,testInteriorBasis->getCardinality());
//          DofOrderingPtr oneVarOrderingTest = Teuchos::rcp(new DofOrdering(testInteriorBasis->domainTopology()));
                    DofOrderingPtr oneVarOrderingTest = Teuchos::rcp(new DofOrdering);
                    oneVarOrderingTest->addEntry(v->ID(), testInteriorBasis, testInteriorBasis->rangeRank());

                    LinearTermPtr f_v = Teuchos::rcp( new LinearTerm );
                    typedef pair< FunctionPtr, VarPtr > LinearSummand;
                    vector<LinearSummand> summands = f->summands();
                    for (int i=0; i<summands.size(); i++) {
                        FunctionPtr f = summands[i].first;
                        if (v->ID() == summands[i].second->ID()) {
                            f_v->addTerm(f * v);
                            f_v->integrate(fValues, oneVarOrderingTest, basisCache);
                        }
                    }

                    LinearTermPtr v_lt = 1.0 * v;
                    FieldContainer<double> l2(numCells,testInteriorBasis->getCardinality(),testInteriorBasis->getCardinality());
                    v_lt->integrate(l2,oneVarOrderingTest,v_lt,oneVarOrderingTest,basisCache,basisCache->isSideCache());

                    Teuchos::Array<int> testTestDim(2), testOneDim(2);
                    testTestDim[0] = testInteriorBasis->getCardinality();
                    testTestDim[1] = testInteriorBasis->getCardinality();
                    testOneDim[0] = testInteriorBasis->getCardinality();
                    testOneDim[1] = 1;
                    FieldContainer<double> projection(testOneDim);
                    for (int cellOrdinal=0; cellOrdinal<numCells; cellOrdinal++) {
                        FieldContainer<double> l2cell(testTestDim,&l2(cellOrdinal,0,0));
                        FieldContainer<double> f_cell(testOneDim,&fValues(cellOrdinal,0));

                        SerialDenseWrapper::solveSystemUsingQR(projection, l2cell, f_cell);

                        // rows in projection correspond to Ae_i, columns to the e_j.  I.e. projection coefficients for e_i are found in the ith row
                        for (int basisOrdinal_j=0; basisOrdinal_j<projection.dimension(0); basisOrdinal_j++) {
                            int testIndex = testOrderInterior->getDofIndex(v->ID(), basisOrdinal_j);
                            rhsVectorTest(cellOrdinal,testIndex) = projection(basisOrdinal_j,0);
                        }
                    }
                }
            }
        }

        // project strong operator applied to field terms, and use this to populate the top left portion of stiffness matrix:
        {
            FieldContainer<double> trialFieldTestInterior(numCells, fieldOrder->totalDofs(), testOrderInterior->totalDofs());
            for (int eqn=0; eqn<numEquations; eqn++) {
                LinearTermPtr Au = _virtualTerms.getFieldOperators()[eqn];
                VarPtr v = _virtualTerms.getFieldTestVars()[eqn];
                set<int> fieldIDs = Au->varIDs();
                for (set<int>::iterator fieldIt = fieldIDs.begin(); fieldIt != fieldIDs.end(); fieldIt++) {
                    int fieldID = *fieldIt;
//          int testID = fieldTestMap[fieldID];
//
//          LinearTermPtr testInteriorVar = 1.0 * this->varFactory().test(testID);

                    BasisPtr vBasis = testOrderInterior->getBasis(v->ID());
                    BasisPtr fieldTrialBasis = fieldOrder->getBasis(fieldID);
//          DofOrderingPtr oneVarOrderingTest = Teuchos::rcp(new DofOrdering(vBasis->domainTopology()));
                    DofOrderingPtr oneVarOrderingTest = Teuchos::rcp(new DofOrdering());
                    oneVarOrderingTest->addEntry(v->ID(), vBasis, vBasis->rangeRank());
                    FieldContainer<double> Au_values(numCells,vBasis->getCardinality(),fieldTrialBasis->getCardinality());
                    FieldContainer<double> l2(numCells,vBasis->getCardinality(),vBasis->getCardinality());

                    DofOrderingPtr oneVarOrderingTrial = Teuchos::rcp(new DofOrdering());
//          DofOrderingPtr oneVarOrderingTrial = Teuchos::rcp(new DofOrdering(fieldTrialBasis->domainTopology()));
                    oneVarOrderingTrial->addEntry(fieldID, fieldTrialBasis, fieldTrialBasis->rangeRank());

                    LinearTermPtr Au_restricted_to_field = Teuchos::rcp( new LinearTerm );
                    typedef pair< FunctionPtr, VarPtr > LinearSummand;
                    vector<LinearSummand> summands = Au->summands();
                    for (int i=0; i<summands.size(); i++) {
                        FunctionPtr f = summands[i].first;
                        VarPtr v = summands[i].second;
                        if (v->ID() == fieldID) {
                            Au_restricted_to_field->addTerm(f * v);
                        }
                    }

                    LinearTermPtr v_lt = 1.0 * v;

                    Au_restricted_to_field->integrate(Au_values,oneVarOrderingTrial,v_lt,oneVarOrderingTest,basisCache,basisCache->isSideCache());
                    v_lt->integrate(l2,oneVarOrderingTest,v_lt,oneVarOrderingTest,basisCache,basisCache->isSideCache());
                    double maxValue = 0;
                    for (int i=0; i<l2.size(); i++) {
                        maxValue = max(abs(l2[i]),maxValue);
                    }
                    cout << "maxValue in l2 is " << maxValue << endl;
                    Teuchos::Array<int> testTestDim(2), trialTestDim(2);
                    testTestDim[0] = vBasis->getCardinality();
                    testTestDim[1] = vBasis->getCardinality();
                    trialTestDim[0] = vBasis->getCardinality();
                    trialTestDim[1] = fieldTrialBasis->getCardinality();
                    FieldContainer<double> projection(trialTestDim);
                    for (int cellOrdinal=0; cellOrdinal<numCells; cellOrdinal++) {
                        FieldContainer<double> l2cell(testTestDim,&l2(cellOrdinal,0,0));
                        FieldContainer<double> AuCell(trialTestDim,&Au_values(cellOrdinal,0,0));
                        // TODO: confirm that I'm doing the right projection here.  I could be missing a key point, but it seems to me that we must
                        //       project onto an *orthonormal* basis here, to achieve the required identity structure of the (field,field) part of the
                        //       Gram matrix.  OTOH, it looks to me like the computation here achieves exactly that, even though I didn't initially
                        //       have that in mind...
//            SerialDenseWrapper::solveSystemUsingQR(projection, l2cell, AuCell);
                        SerialDenseWrapper::solveSystemMultipleRHS(projection, l2cell, AuCell);

                        // rows in projection correspond to Ae_i, columns to the e_j.  I.e. projection coefficients for e_i are found in the ith row
                        for (int basisOrdinal_i=0; basisOrdinal_i<projection.dimension(0); basisOrdinal_i++) {
                            int testIndex = testOrderInterior->getDofIndex(v->ID(), basisOrdinal_i);
                            for (int basisOrdinal_j=0; basisOrdinal_j<projection.dimension(1); basisOrdinal_j++) {
                                int trialIndex = fieldOrder->getDofIndex(fieldID, basisOrdinal_j); // in the *trial* space
                                trialFieldTestInterior(cellOrdinal,trialIndex,testIndex) = projection(basisOrdinal_i,basisOrdinal_j);
                            }
                        }
                    }
                }
            }
            Teuchos::Array<int> trialTestDim(2);
            trialTestDim[0] = fieldOrder->totalDofs();
            trialTestDim[1] = testOrderInterior->totalDofs();
            for (int cellOrdinal=0; cellOrdinal<numCells; cellOrdinal++) {
                FieldContainer<double> trialFieldTrialField(fieldOrder->totalDofs(), fieldOrder->totalDofs());
                FieldContainer<double> trialTestCell(trialTestDim, &trialFieldTestInterior(cellOrdinal,0,0));
                SerialDenseWrapper::multiply(trialFieldTrialField, trialTestCell, trialTestCell, 'N', 'T'); // transpose the second one
                // now, accumulate into localStiffness
                for (int i=0; i<trialFieldTrialField.dimension(0); i++) {
                    int stiff_i = stiffnessIndexForFieldIndex[i];
                    for (int j=0; j<trialFieldTrialField.dimension(1); j++) {
                        int stiff_j = stiffnessIndexForFieldIndex[j];
                        localStiffness(cellOrdinal,stiff_i,stiff_j) = trialFieldTrialField(i,j);
                    }
                }
            }
            // multiply RHS integrated against the interior test space by the trialFieldTestInterior
            for (int cellOrdinal=0; cellOrdinal<numCells; cellOrdinal++) {
                Teuchos::Array<int> trialTestDim(2), oneTestDim(2);
                trialTestDim[0] = fieldOrder->totalDofs();
                trialTestDim[1] = testOrderInterior->totalDofs();
                oneTestDim[0] = 1;
                oneTestDim[1] = testOrderInterior->totalDofs();
                FieldContainer<double> trialTestCell(trialTestDim, &trialFieldTestInterior(cellOrdinal,0,0));
                FieldContainer<double> rhsTestCell(oneTestDim, &rhsVectorTest(cellOrdinal,0));
                FieldContainer<double> rhsTrialCell(1, fieldOrder->totalDofs());

                SerialDenseWrapper::multiply(rhsTrialCell, rhsTestCell, trialTestCell, 'N', 'T');

                for (int fieldIndex=0; fieldIndex<fieldOrder->totalDofs(); fieldIndex++) {
                    int stiffIndex = stiffnessIndexForFieldIndex[fieldIndex];
                    rhsVector(cellOrdinal,stiffIndex) = rhsTrialCell(0, fieldIndex);
                }
            }
        }

        FieldContainer<double> ipMatrixTraceIncludingInterior(numCells,numTestTraceDofsIncludingInterior,numTestTraceDofsIncludingInterior);
        int numTestTerms = _virtualTerms.getTestNormOperators().size();
        for (int i=0; i<numTestTerms; i++) {
            LinearTermPtr testTerm = _virtualTerms.getTestNormOperators()[i];
            LinearTermPtr boundaryTerm = _virtualTerms.getTestNormBoundaryOperators()[i];
            testTerm->integrate(ipMatrixTraceIncludingInterior,testOrderTrace,boundaryTerm,testOrderTrace,ipBasisCache,ipBasisCache->isSideCache());
        }
        FieldContainer<double> ipMatrixTrace(numCells,numTestTraceDofs,numTestTraceDofs);
        for (int cellOrdinal=0; cellOrdinal<numCells; cellOrdinal++) {
            for (int i_dofIndex=0; i_dofIndex<numTestTraceDofsIncludingInterior; i_dofIndex++) {
                if (remappedTraceIndices.find(i_dofIndex) == remappedTraceIndices.end()) {
                    continue;
                }
                int i_remapped = remappedTraceIndices[i_dofIndex];
                for (int j_dofIndex=0; j_dofIndex<numTestTraceDofsIncludingInterior; j_dofIndex++) {
                    if (remappedTraceIndices.find(j_dofIndex) == remappedTraceIndices.end()) {
                        continue;
                    }
                    int j_remapped = remappedTraceIndices[j_dofIndex];
                    ipMatrixTrace(cellOrdinal,i_remapped,j_remapped) = ipMatrixTraceIncludingInterior(cellOrdinal,i_dofIndex,j_dofIndex);
                }
            }
        }

        testMatrixAssemblyTime += timer.ElapsedTime();
        //      cout << "ipMatrix:\n" << ipMatrix;

        timer.ResetStartTime();

        cout << "NOTE: we do not yet enforce continuity on the trace test space.\n"; // I *think* this is fine, but I'm not dead certain -- we do of course in the end enforce continuity in GDAMinimumRule

        // now, determine the trace part of the bilinear form matrix
        FieldContainer<double> bfMatrixTraceTraceIncludingTestInterior(numCells,testOrderTrace->totalDofs(),traceOrder->totalDofs());
        FieldContainer<double> bfMatrixFieldTraceIncludingTestInterior(numCells,testOrderTrace->totalDofs(),fieldOrder->totalDofs());
        for (int eqn=0; eqn<numEquations; eqn++) {
            VarPtr traceVar = _virtualTerms.getTraceVars()[eqn];
            LinearTermPtr termTraced = traceVar->termTraced();
            LinearTermPtr strongOperator = _virtualTerms.getFieldOperators()[eqn];
            VarPtr testVar = _virtualTerms.getFieldTestVars()[eqn];

            // want to determine \hat{C}(\hat{e}_i, \phi_j) for \phi_j with support on the boundary
            // the \phi_j's with support on the boundary are the ones associated with the trace

            LinearTermPtr trialTerm = 1.0 * traceVar;
            LinearTermPtr testTerm;

            if (traceVar->varType() == TRACE) {
                testTerm = Function::normal() * testVar;
            } else {
                testTerm = 1.0 * testVar;
            }

//      trialTerm->integrate(bfMatrixTrace,traceOrder,testTerm,testOrderTrace,basisCache,basisCache->isSideCache());
            trialTerm->integrate(bfMatrixTraceTraceIncludingTestInterior,traceOrder,testTerm,testOrderTrace,basisCache,basisCache->isSideCache());
            termTraced->integrate(bfMatrixFieldTraceIncludingTestInterior,fieldOrder,-testTerm,testOrderTrace,basisCache,basisCache->isSideCache());
        }

        FieldContainer<double> bfMatrixFieldTrace(numCells,numTestTraceDofs,bfMatrixFieldTraceIncludingTestInterior.dimension(2));
        FieldContainer<double> bfMatrixTraceTrace(numCells,numTestTraceDofs,bfMatrixTraceTraceIncludingTestInterior.dimension(2));
        for (int cellOrdinal=0; cellOrdinal<numCells; cellOrdinal++) {
            for (int i_dofIndex=0; i_dofIndex<numTestTraceDofsIncludingInterior; i_dofIndex++) {
                if (remappedTraceIndices.find(i_dofIndex) == remappedTraceIndices.end()) {
                    continue;
                }
                int i_remapped = remappedTraceIndices[i_dofIndex];
                for (int j_dofIndex=0; j_dofIndex<bfMatrixFieldTrace.dimension(2); j_dofIndex++) {
                    bfMatrixFieldTrace(cellOrdinal,i_remapped,j_dofIndex) = bfMatrixFieldTraceIncludingTestInterior(cellOrdinal,i_dofIndex,j_dofIndex);
                }
                for (int j_dofIndex=0; j_dofIndex<bfMatrixTraceTrace.dimension(2); j_dofIndex++) {
                    bfMatrixTraceTrace(cellOrdinal,i_remapped,j_dofIndex) = bfMatrixTraceTraceIncludingTestInterior(cellOrdinal,i_dofIndex,j_dofIndex);
                }
            }
        }

        Teuchos::Array<int> ipMatrixDim(2), bfMatrixTraceTraceDim(2), bfMatrixFieldTraceDim(2);
        Teuchos::Array<int> traceTraceStiffDim(2), fieldTraceStiffDim(2), fieldFieldStiffDim(2);
        ipMatrixDim[0] = ipMatrixTrace.dimension(1);
        ipMatrixDim[1] = ipMatrixTrace.dimension(2);

        bfMatrixTraceTraceDim[0] = bfMatrixTraceTrace.dimension(1);
        bfMatrixTraceTraceDim[1] = bfMatrixTraceTrace.dimension(2);

        bfMatrixFieldTraceDim[0] = bfMatrixFieldTrace.dimension(1);
        bfMatrixFieldTraceDim[1] = bfMatrixFieldTrace.dimension(2);

        traceTraceStiffDim[0] = traceOrder->totalDofs();
        traceTraceStiffDim[1] = traceTraceStiffDim[0];

        fieldTraceStiffDim[0] = fieldOrder->totalDofs();
        fieldTraceStiffDim[1] = traceOrder->totalDofs(); // rectangular

        fieldFieldStiffDim[0] = fieldOrder->totalDofs();
        fieldFieldStiffDim[1] = fieldOrder->totalDofs();

        FieldContainer<double> traceTraceStiffCell(traceTraceStiffDim);
        FieldContainer<double> fieldTraceStiffCell(fieldTraceStiffDim);
        FieldContainer<double> fieldFieldStiffCell(fieldFieldStiffDim);
        for (int cellOrdinal=0; cellOrdinal<numCells; cellOrdinal++) {
            FieldContainer<double> ipMatrixCell(ipMatrixDim,&ipMatrixTrace(cellOrdinal,0,0));

            FieldContainer<double> optTestCoeffsTraceTrace(numTestTraceDofs,traceOrder->totalDofs());
            FieldContainer<double> bfMatrixTraceTraceCell(bfMatrixTraceTraceDim,&bfMatrixTraceTrace(cellOrdinal,0,0));
            int result = SerialDenseWrapper::solveSystemUsingQR(optTestCoeffsTraceTrace, ipMatrixCell, bfMatrixTraceTraceCell);
            SerialDenseWrapper::multiply(traceTraceStiffCell, bfMatrixTraceTraceCell, optTestCoeffsTraceTrace, 'T', 'N');

            // copy into the appropriate spot in localStiffness:
            for (int i=0; i<traceTraceStiffDim[0]; i++) {
                int i_stiff = stiffnessIndexForTraceIndex[i];
                for (int j=0; j<traceTraceStiffDim[1]; j++) {
                    int j_stiff = stiffnessIndexForTraceIndex[j];
                    localStiffness(cellOrdinal,i_stiff,j_stiff) = traceTraceStiffCell(i,j);
                }
            }

            // because of the way the matrix blocks line up, we actually don't have to do a second inversion of ipMatrixCell for this part
            FieldContainer<double> bfMatrixFieldTraceCell(bfMatrixFieldTraceDim,&bfMatrixFieldTrace(cellOrdinal,0,0));
            SerialDenseWrapper::multiply(fieldTraceStiffCell, bfMatrixFieldTraceCell, optTestCoeffsTraceTrace, 'T', 'N');

            // copy into the appropriate spots in localStiffness (taking advantage of symmetry):
            for (int i=0; i<fieldTraceStiffDim[0]; i++) {
                int i_stiff = stiffnessIndexForFieldIndex[i];
                for (int j=0; j<fieldTraceStiffDim[1]; j++) {
                    int j_stiff = stiffnessIndexForTraceIndex[j];
                    localStiffness(cellOrdinal,i_stiff,j_stiff) = fieldTraceStiffCell(i,j);
                    localStiffness(cellOrdinal,j_stiff,i_stiff) = fieldTraceStiffCell(i,j);
                }
            }

            // because of the way the matrix blocks line up, we do have some trace contributions in the (field, field) portion of the matrix
            // these get added to the field contributions (hence the +=)
            FieldContainer<double> optTestCoeffsFieldTrace(numTestTraceDofs,fieldOrder->totalDofs());
            result = SerialDenseWrapper::solveSystemUsingQR(optTestCoeffsFieldTrace, ipMatrixCell, bfMatrixFieldTraceCell);
            SerialDenseWrapper::multiply(fieldFieldStiffCell, bfMatrixFieldTraceCell, optTestCoeffsFieldTrace, 'T', 'N');
            for (int i=0; i<fieldFieldStiffDim[0]; i++) {
                int i_stiff = stiffnessIndexForFieldIndex[i];
                for (int j=0; j<fieldFieldStiffDim[1]; j++) {
                    int j_stiff = stiffnessIndexForFieldIndex[j];
                    localStiffness(cellOrdinal,i_stiff,j_stiff) += fieldFieldStiffCell(i,j);
                }
            }
        }

        testMatrixInversionTime += timer.ElapsedTime();
        //      cout << "optTestCoeffs:\n" << optTestCoeffs;

        if (printTimings) {
            cout << "testMatrixAssemblyTime: " << testMatrixAssemblyTime << " seconds.\n";
            cout << "testMatrixInversionTime: " << testMatrixInversionTime << " seconds.\n";
            cout << "localStiffnessDeterminationFromTestsTime: " << localStiffnessDeterminationFromTestsTime << " seconds.\n";
        }
    }