vector<double> FEM::computeDiscreteSolution(DiffusionReactionEqn const & PDE,
Triangulation& Omega,
BoundaryConditions& BCs) {
// data structures for final linear system A.xi = b:
SymmetricCSlRMatrix A(Omega.generateAdjList()); // build final matrix portrait
vector<double> b(Omega.numbOfNodes(), 0.), // load vector
xi(Omega.numbOfNodes(), 0.); // discrete solution
// data structures for assemby of A and b:
SymmetricContainer<double> localMassMatrix(3), // for hat functions on triangles
localStiffnessMatrix(3), // we have 3 × 3 element matricies
localRobinMatrix(2); // and 2 × 2 element matricies for Robin BCs (just like element matrix in 1D)
array<double, 3> localLoadVector; // and their
array<double, 2> localRobinVector; // friends, element vectors
array<Node, 3> elementNodes, // nodes of the current triangle
elementMiddleNodes; // and nodes on the middle of edges
array<Node, 2> edgeNodes; // nodes spanning an edge of the current triangle that is part of bndry
Node midPoint; // middle point of the edge (to define which BCs to apply)
double measure; // area of ith triangle / length of bndry edge of ith thiangle
array<size_t, 3> l2g_elem; // local to global mapping of nodes on the element
array<size_t, 2> l2g_edge; // and on the edge
LocalIndex j, k, leftNodeIndex, rightNodeIndex; // dummy indicies
for (size_t i = 0; i < Omega.numbOfTriangles(); ++i) {
// (1) quadratures over elements
// in order to assemble stiffness matrix and load vector,
// it is convenient to iterate over mesh elements (i.e. triangles)
elementNodes = Omega.getNodes(i); // get nodes of ith triangle
for (j = 0; j < 3; ++j) // and middle nodes of its edges
elementMiddleNodes[j] = elementNodes[k = nextIndex(j)].midPoint(elementNodes[nextIndex(k)]);
measure = Omega.area(i); // compute area of ith triangle
l2g_elem = Omega.l2g(i); // local to global mapping of nodes of ith element
// compute
// (1.1) local mass matrix,
// (1.2) local stiffness matrix, and
// (1.3) local load vector
localStiffnessMatrix = computeLocalStiffnessMatrix(PDE.diffusionTerm(), elementNodes, elementMiddleNodes, measure);
localMassMatrix = computeLocalMassMatrix(PDE.reactionTerm(), elementNodes, measure);
localLoadVector = computeLocalLoadVector(PDE.forceTerm(), elementNodes, elementMiddleNodes, measure);
// (1.4) assemble contributions
for (j = 0; j < 3; ++j) {
for (k = j; k < 3; ++k)
A(l2g_elem[j], l2g_elem[k]) += localMassMatrix(j, k) + localStiffnessMatrix(j, k);
b[l2g_elem[j]] += localLoadVector[j];
}
// (2) quadratures over edges
// iterate over list of local indicies of boundary nodes
for (LocalIndex edgeIndex : Omega.getBoundaryIndicies(i)) {
// if edgeIndex = 2, then the edge against second node of ith triangle
// is part of the boundary
// so we need to assemble BCs here
leftNodeIndex = nextIndex(edgeIndex); // local indicies of nodes that
rightNodeIndex = nextIndex(leftNodeIndex); // define the edge
edgeNodes = { elementNodes[leftNodeIndex], elementNodes[rightNodeIndex] }; // and the nodes themselves
l2g_edge[0] = l2g_elem[leftNodeIndex]; // local to global nodes
l2g_edge[1] = l2g_elem[rightNodeIndex]; // numeration mapping
measure = Omega.length(i, edgeIndex);
// define BCs to apply
midPoint = edgeNodes[0].midPoint(edgeNodes[1]);
BCs.defineBCsAt(midPoint);
// compute
// (2.1) local Robin matrix
// (2.2) local Robin vector
localRobinMatrix = computeLocalRobinMatrix(BCs, edgeNodes, measure);
localRobinVector = computeLocalRobinVector(BCs, edgeNodes, measure);
// (2.3) assemble contributions
for (j = 0; j < 2; ++j) {
for (k = j; k < 2; ++k)
A(l2g_edge[j], l2g_edge[k]) += localRobinMatrix(j, k);
b[l2g_edge[j]] += localRobinVector[j];
}
}
}
// now we are ready to compute xi, A.xi = b
xi = CG(A, b, xi, 10e-70);
return xi;
}
vector<double> FEM::constructVector(Function u, Triangulation& Omega) {
vector<double> uVec(Omega.numbOfNodes());
for (size_t i = 0; i < uVec.size(); ++i)
uVec[i] = u(Omega.getNode(i));
return uVec;
}
