Piecewise linear space with nodal basis functions on a two-dimensional triangular mesh. More...

Data Structures
struct	_tri2dp1
	Representation of a trial space with piecewise linear nodal basis functions on a two-dimensional triangular mesh. More...

Typedefs
typedef struct _tri2dp1	tri2dp1
	Representation of a trial space with piecewise linear nodal basis functions on a two-dimensional triangular mesh.

typedef tri2dp1 *	ptri2dp1
	Pointer to a tri2p1 object.

typedef const tri2dp1 *	pctri2dp1
	Pointer to a constant tri2p1 object.

Functions
ptri2dp1	new_tri2dp1 (pctri2d gr)
	Create a tri2p1 object using a tri2d mesh. More...

void	del_tri2dp1 (ptri2dp1 dc)
	Delete a tri2p1 object. More...

psparsematrix	build_tri2dp1_sparsematrix (pctri2dp1 dc)
	Create a sparsematrix with a sparsity pattern matching the nodal basis functions representing the system matrix. More...

psparsematrix	build_tri2dp1_interaction_sparsematrix (pctri2dp1 dc)
	Create a sparsematrix with a sparsity pattern matching the nodal basis functions representing the interaction matrix. More...

psparsematrix	build_tri2dp1_prolongation_sparsematrix (pctri2dp1 dfine, pctri2dp1 dcoarse, pctri2dref rf)
	Create a prolongation matrix. More...

void	assemble_tri2dp1_laplace_sparsematrix (pctri2dp1 dc, psparsematrix A, psparsematrix Af)
	Assemble stiffness matrix. More...

void	assemble_tri2dp1_mass_sparsematrix (pctri2dp1 dc, psparsematrix M, psparsematrix Mf)
	Assemble mass matrix. More...

void	assemble_tri2dp1_dirichlet_avector (pctri2dp1 dc, field(d)(const real x, void fdata), void fdata, pavector dv)
	Discretize Dirichlet boundary values. More...

void	assemble_tri2dp1_functional_avector (ptri2dp1 dc, field(f)(const real x, void fdata), void fdata, pavector fv)
	Discretize an -functional. More...

real	normmax_tri2dp1 (pctri2dp1 dc, field(u)(const real x, void fdata), void fdata, pavector xs)
	Compute the vertex-wise maximum norm of the discretization error. More...

real	norml2_tri2dp1 (pctri2dp1 dc, field(f)(const real x, void fdata), void fdata, pcavector xs, pcavector xf)
	Compute the -norm of the discretization error. More...

pclustergeometry	build_tri2dp1_clustergeometry (pctri2dp1 p1, uint *idx)
	Create a clustergeometry object for a FEM-discretisation with p1-functions. More...

Detailed Description

Piecewise linear space with nodal basis functions on a two-dimensional triangular mesh.

Function Documentation

void assemble_tri2dp1_dirichlet_avector	(	pctri2dp1	dc,
		field()(const real x, void *fdata)	d,
		void *	fdata,
		pavector	dv
	)

Discretize Dirichlet boundary values.

For each fixed vertex, the function $f$ is evaluated and the result is written to the corresponding entry in the result. In the standard case, this provides us with a coefficient vector corresponding to the piecewise linear interpolation of the Dirichlet boundary values. Subtracting $A_f d$ from the right-hand side of our equation, where $A_f$ is the interaction matrix, yields the right-hand side of the problem with homogeneous Dirichlet boundary conditions.

Parameters

dc	tri2p1 object describing the trial space.
d	Dirichlet boundary values.
fdata	Additional data for `f`.
dv	Target vector of dimension `dc->nfix`.

void assemble_tri2dp1_functional_avector	(	ptri2dp1	dc,
		field()(const real x, void *fdata)	f,
		void *	fdata,
		pavector	fv
	)

Discretize an $L^2$ -functional.

For each triangle, the element vector is computed by applying the edge-midpoint quadrature. Its components are then added to the corresponding entries in $b$ . Since $b$ is cleared before this step, the result is given by $b_i = \int f(x) \varphi_i(x) \,dx$ .

Parameters

dc	tri2p1 object describing the trial space.
f	Right-hand side function.
fdata	Additional data for `f`.
fv	Target vector of dimension `dc->ndof`.

void assemble_tri2dp1_laplace_sparsematrix	(	pctri2dp1	dc,
		psparsematrix	A,
		psparsematrix	Af
	)

Assemble stiffness matrix.

Element matrices for all triangles are computed and added to the corresponding entries in A and Af.

Parameters

dc	tri2p1 object describing the trial space.
A	Target matrix for degrees of freedom.
Af	Target matrix for interactions between fixed vertices and degrees of freedom.

void assemble_tri2dp1_mass_sparsematrix	(	pctri2dp1	dc,
		psparsematrix	M,
		psparsematrix	Mf
	)

Assemble mass matrix.

Element matrices for all triangles are computed and added to the corresponding entries in M and Mf.

Parameters

dc	tri2p1 object describing the trial space.
M	Target matrix for degrees of freedom.
Mf	Target matrix for interactions between fixed vertices and degrees of freedom.

pclustergeometry build_tri2dp1_clustergeometry	(	pctri2dp1	p1,
		uint *	idx
	)

Create a clustergeometry object for a FEM-discretisation with p1-functions.

Create a clustergeometry object basing on a triangular tri2d mesh and linear nodal basis function.

Parameters

p1	tri2dp1 object.
idx	Array of indices.

Returns: Clustergeometry object.

psparsematrix build_tri2dp1_interaction_sparsematrix ( pctri2dp1 dc )

Create a sparsematrix with a sparsity pattern matching the nodal basis functions representing the interaction matrix.

Creates a sparse matrix matching the nodal basis functions: if the supports of basis functions $\varphi_i$ and $\varphi_j$ share at least one edge, the entry $a_{ij}$ appears in the graph of the matrix.

Here, only $i$ refers to a degree of freedom, while $j$ is the index of a fixed vertex, i.e., has to be between 0 and nfix-1.

Parameters

dc	tri2p1 object describing the space.

Returns: Sparse matrix with dc->ndof rows and dc->nfix columns.

psparsematrix build_tri2dp1_prolongation_sparsematrix	(	pctri2dp1	dfine,
		pctri2dp1	dcoarse,
		pctri2dref	rf
	)

Create a prolongation matrix.

dcoarse and dfine have to correspond to a coarse mesh and its refinement created by refine_tri2d. The refinement relationship is described by rf.

This function not only creates the matrix, but also fills it with correct coefficients.

Parameters

dfine	tri2p1 object corresponding to the refined mesh.
dcoarse	tri2p1 object corresponding to the coarse mesh.
rf	Refinement relationship.

Returns: sparsematrix containing the prolongation matrix.

psparsematrix build_tri2dp1_sparsematrix ( pctri2dp1 dc )

Create a sparsematrix with a sparsity pattern matching the nodal basis functions representing the system matrix.

Creates a sparse matrix matching the nodal basis functions: if the supports of basis functions $\varphi_i$ and $\varphi_j$ share at least one edge, the entry $a_{ij}$ appears in the graph of the matrix.

Here, both $i$ and $j$ refer to degrees of freedom, i.e., there indices are between 0 and ndof-1.

Parameters

dc	tri2p1 object describing the space.

Returns: Sparse matrix with dc->ndof rows and columns.

void del_tri2dp1 ( ptri2dp1 dc )

Delete a tri2p1 object.

Parameters

dc	Object to be deleted.

ptri2dp1 new_tri2dp1 ( pctri2d gr )

Create a tri2p1 object using a tri2d mesh.

The boundary flags gr->xb are used to decide whether a vertex corresponds to a degree of freedom or is fixed.

Parameters

gr Mesh

Returns: tri2p1 object matching gr.

real norml2_tri2dp1	(	pctri2dp1	dc,
		field()(const real x, void *fdata)	f,
		void *	fdata,
		pcavector	xs,
		pcavector	xf
	)

Compute the $L^2$ -norm of the discretization error.

Compare the function f to the discrete function described by the vector xs (for the degrees of freedom) and xf (for the fixed vertices). Measure the difference in the $L^2$ -norm.

Parameters

dc	tri2p1 object describing the trial space.
f	Function.
fdata	Additional data for `f`.
xs	Coefficients of discrete function.
xf	Coefficients of fixed part of discrete function

Returns: -norm of discretization error.

real normmax_tri2dp1	(	pctri2dp1	dc,
		field()(const real x, void *fdata)	u,
		void *	fdata,
		pavector	xs
	)

Compute the vertex-wise maximum norm of the discretization error.

Evaluate f in all non-fixed vertices and compute the difference with the corresponding component of xs. The maximum of the absolute values gives the vertex-wise $\ell^\infty$ -norm of the discretization error.

Parameters

dc	tri2p1 object describing the trial space.
u	Function.
fdata	Additional data for `f`.
xs	Coefficients of discrete function.

Returns: $\ell^\infty$ -norm of discretization error.

Data Structures

Typedefs

Functions

Detailed Description

Function Documentation