Discusses the assembly the matrix for the bilinear form the left-hand-side

SOLUTION of (3-8.c):
[Lecture → Ex. 2.7.4.37] discusses the assembly of the S0 1(M)-Galerkin matrix for the bilinear form on the left-hand-side of (3.8.3). Of course, for the function solveImpedanceBVP() only the edges on Γ0 should be visited by the Eval() method of .

Of course, we also have to take care of the non-zero Dirichlet boundary conditions on Γg, as demonstrated in [Lecture → Code 2.7.6.17]; use the function with a SELECTOR argument that singles out edges ong. To that end query the position of the edge and rely on the information Ω0 ⊂ [0.1, 0.9]2.

Eigen : : VectorXd solveImpedanceBVP(
const std : : shared_ptr< l f : : uscalfe : : FeSpaceLagrangeO1<double>> &fe_space_p ,
Eigen : : Vector2d g ) {
// Related implementations:
// Homework problem ErrorEstimatesForTraces:
//
https://gitlab.math.ethz.ch/ralfh/npdecodes/tree/master/homeworks/ErrorEstimatesFor

// Pointer to current mesh
std : : shared_ptr<const l f : : mesh : : Mesh> mesh_p = fe_space_p −>Mesh ( ) ; // Obtain local->global index mapping for current finite element space const l f : : assemble : : DofHandler &dofh { fe_space_p −>LocGlobMap ( ) } ;
// Dimension of finite element space
const
l f : : uscalfe : : size_type N_dofs ( dofh . NumDofs ( ) ) ; // Obtain specification for shape functions on edges
const l f : : fe : : ScalarReferenceFiniteElement <double> * rsf_edge_p = fe_space_p −>ShapeFunctionLayout ( l f : : base : : RefEl : : kSegment ( ) ) ;

// I.i : Computing volume matrix for negative Laplace operator
// Initialize object taking care of local mass (volume) computations.

l f : : uscalfe : : LinearFELaplaceElementMatrix elmat_builder { } ;
// Invoke assembly on cells (co-dimension = 0 as first argument)
// Information about the mesh and the local-to-global map is passed through // a Dofhandler object, argument ’dofh’. This function call adds triplets to // the internal COO-format representation of the sparse matrix A.

	dofh ,	dofh ,

// I.ii : Computing mass edge matrix resulting from Robin B.C.

[& bd_flags ] ( const
		{
auto endpoints =		l f : : geometry : : Corners ( * ( edge . Geometry ( ) ) ) ;
i f	( endpoints (0 ,	0) <= 0.05	\| \|	0.95 <= endpoints (0 ,	0)
endpoints (1 ,		0) <= 0.05	\| \|	0.95 <= endpoints (1 ,	0) )
return		0) <= 0.05	\| \|	0.95 <= endpoints (1 ,	0) )

edgemat_builder ( fe_space_p ,	eta ,	edges_predicate_RobinBC ) ;

	dofh ,	dofh ,

Function solveImpedanceBVP: Assembly of right-hand-side vector and

auto mf_f = l f : : mesh : : u t i l s : : MeshFunctionGlobal (

[ ] ( Eigen : : Vector2d x ) −> double { return 0.0; } ) ;

	dofh ,	elvec_builder ,	phi ) ;

// I.iv : Imposing essential boundary conditions

// InitEssentialConditionFromFunction()

//

[& bd_flags ] ( const			l f : : mesh : : E n t i t y &edge ) −> bool {
( bd_flags ( edge ) )
auto endpoints =
i f	( endpoints (0 ,		0) <= 0.05	\| \|	0.95 <= endpoints (0 ,	0)
endpoints (1 ,			0) <= 0.05	\| \|	0.95 <= endpoints (1 ,	0) )
return		true ;	0) <= 0.05	\| \|	0.95 <= endpoints (1 ,	0) )