#include <staticcond.hpp>

Public Member Functions
	StaticCondensation (FiniteElementSpace *fespace)
	Construct a StaticCondensation object. More...

	~StaticCondensation ()
	Destroy a StaticCondensation object. More...

int	GetNPrDofs () const
	Return the number of vector private dofs. More...

int	GetNExDofs () const
	Return the number of vector exposed/reduced dofs. More...

bool	ReducesTrueVSize () const

void	Init (bool symmetric, bool block_diagonal)

FiniteElementSpace *	GetTraceFESpace ()
	Return a pointer to the reduced/trace FE space. More...

ParFiniteElementSpace *	GetParTraceFESpace ()
	Return a pointer to the parallel reduced/trace FE space. More...

void	AssembleMatrix (int el, const DenseMatrix &elmat)

void	AssembleBdrMatrix (int el, const DenseMatrix &elmat)

void	Finalize ()
	Finalize the construction of the Schur complement matrix. More...

void	EliminateReducedTrueDofs (const Array< int > &ess_rtdof_list, int keep_diagonal)
	Eliminate the given reduced true dofs from the Schur complement matrix S. More...

bool	HasEliminatedBC () const

SparseMatrix &	GetMatrix ()
	Return the serial Schur complement matrix. More...

SparseMatrix &	GetMatrixElim ()
	Return the eliminated part of the serial Schur complement matrix. More...

HypreParMatrix &	GetParallelMatrix ()
	Return the parallel Schur complement matrix. More...

HypreParMatrix &	GetParallelMatrixElim ()
	Return the eliminated part of the parallel Schur complement matrix. More...

void	ReduceRHS (const Vector &b, Vector &sc_b) const

void	ReduceSolution (const Vector &sol, Vector &sc_sol) const

void	ConvertMarkerToReducedTrueDofs (const Array< int > &ess_tdof_marker, Array< int > &ess_rtdof_marker) const

void	ConvertListToReducedTrueDofs (const Array< int > &ess_tdof_list, Array< int > &ess_rtdof_list) const

void	ComputeSolution (const Vector &b, const Vector &sc_sol, Vector &sol) const

Detailed Description

Auxiliary class StaticCondensation, used to implement static condensation in class BilinearForm.

Static condensation is a technique for solving linear systems by eliminating groups/blocks of unknowns and reducing the original system to the remaining interfacial unknowns. The assumption is that unknowns in one group are connected (in the graph of the matrix) only to unknowns in the same group or to interfacial unknowns but not to other groups.

For finite element systems, the groups correspond to degrees of freedom (DOFs) associated with the interior of the elements. The rest of the DOFs (associated with the element boundaries) are interfacial.

In block form the matrix of the system can be written as

$A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \begin{array}{l} \text{-- groups: element interior/private DOFs} \\ \text{-- interface: element boundary/exposed DOFs} \end{array}$

where the block $ A_1 $ is itself block diagonal with small local blocks and it is, therefore, easily invertible.

Starting with the block system

$\begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \begin{pmatrix} X_1 \\ X_2 \end{pmatrix} = \begin{pmatrix} B_1 \\ B_2 \end{pmatrix}$

the reduced, statically condensed system is given by

$S_{22} X_2 = B_2 - A_{21} A_{11}^{-1} B_1$

where the Schur complement matrix $S_{22}$ is given by

$S_{22} = A_{22} - A_{21} A_{11}^{-1} A_{12}.$

After solving the Schur complement system, the $ X_1 $ part of the solution can be recovered using the formula

$X_1 = A_{11}^{-1} ( B_1 - A_{12} X_2 ).$

Definition at line 65 of file staticcond.hpp.

Constructor & Destructor Documentation

mfem::StaticCondensation::StaticCondensation ( FiniteElementSpace * fespace )

Construct a StaticCondensation object.

Definition at line 17 of file staticcond.cpp.

mfem::StaticCondensation::~StaticCondensation ( )

Destroy a StaticCondensation object.

Definition at line 104 of file staticcond.cpp.

Member Function Documentation

void mfem::StaticCondensation::AssembleBdrMatrix	(	int	el,
		const DenseMatrix &	elmat
	)

Assemble the contribution to the Schur complement from the given boundary element matrix 'elmat'.

Definition at line 227 of file staticcond.cpp.

void mfem::StaticCondensation::AssembleMatrix	(	int	el,
		const DenseMatrix &	elmat
	)

Assemble the contribution to the Schur complement from the given element matrix 'elmat'; save the other blocks internally: A_pp_inv, A_pe, and A_ep.

Definition at line 189 of file staticcond.cpp.

void mfem::StaticCondensation::ComputeSolution	(	const Vector &	b,
		const Vector &	sc_sol,
		Vector &	sol
	)		const

Given a solution of the reduced system 'sc_sol' and the RHS 'b' for the full linear system, compute the solution of the full system 'sol'.

Definition at line 452 of file staticcond.cpp.

void mfem::StaticCondensation::ConvertListToReducedTrueDofs	(	const Array< int > &	ess_tdof_list,
		Array< int > &	ess_rtdof_list
	)		const

inline

Restrict a list of true FE space dofs to a list of reduced/trace true FE space dofs.

Definition at line 169 of file staticcond.hpp.

void mfem::StaticCondensation::ConvertMarkerToReducedTrueDofs	(	const Array< int > &	ess_tdof_marker,
		Array< int > &	ess_rtdof_marker
	)		const

Restrict a marker Array on the true FE space dofs to a marker Array on the reduced/trace true FE space dofs.

Definition at line 415 of file staticcond.cpp.

void mfem::StaticCondensation::EliminateReducedTrueDofs	(	const Array< int > &	ess_rtdof_list,
		int	keep_diagonal
	)

Eliminate the given reduced true dofs from the Schur complement matrix S.

Definition at line 286 of file staticcond.cpp.

void mfem::StaticCondensation::Finalize ( )

Finalize the construction of the Schur complement matrix.

Definition at line 235 of file staticcond.cpp.

SparseMatrix& mfem::StaticCondensation::GetMatrix ( )

inline

Return the serial Schur complement matrix.

Definition at line 141 of file staticcond.hpp.

SparseMatrix& mfem::StaticCondensation::GetMatrixElim ( )

inline

Return the eliminated part of the serial Schur complement matrix.

Definition at line 144 of file staticcond.hpp.

int mfem::StaticCondensation::GetNExDofs ( ) const

inline

Return the number of vector exposed/reduced dofs.

Definition at line 97 of file staticcond.hpp.

int mfem::StaticCondensation::GetNPrDofs ( ) const

inline

Return the number of vector private dofs.

Definition at line 95 of file staticcond.hpp.

HypreParMatrix& mfem::StaticCondensation::GetParallelMatrix ( )

inline

Return the parallel Schur complement matrix.

Definition at line 148 of file staticcond.hpp.

HypreParMatrix& mfem::StaticCondensation::GetParallelMatrixElim ( )

inline

Return the eliminated part of the parallel Schur complement matrix.

Definition at line 151 of file staticcond.hpp.

ParFiniteElementSpace* mfem::StaticCondensation::GetParTraceFESpace ( )

inline

Return a pointer to the parallel reduced/trace FE space.

Definition at line 111 of file staticcond.hpp.

FiniteElementSpace* mfem::StaticCondensation::GetTraceFESpace ( )

inline

Return a pointer to the reduced/trace FE space.

Definition at line 107 of file staticcond.hpp.

bool mfem::StaticCondensation::HasEliminatedBC ( ) const

inline

Return true if essential boundary conditions have been eliminated from the Schur complement matrix.

Definition at line 131 of file staticcond.hpp.

void mfem::StaticCondensation::Init	(	bool	symmetric,
		bool	block_diagonal
	)

Prepare the StaticCondensation object to assembly: allocate the Schur complement matrix and the other element-wise blocks.

Definition at line 134 of file staticcond.cpp.

void mfem::StaticCondensation::ReduceRHS	(	const Vector &	b,
		Vector &	sc_b
	)		const

Given a RHS vector for the full linear system, compute the RHS for the reduced linear system: sc_b = b_e - A_ep A_pp_inv b_p.

Definition at line 309 of file staticcond.cpp.

void mfem::StaticCondensation::ReduceSolution	(	const Vector &	sol,
		Vector &	sc_sol
	)		const

Restrict a solution vector on the full FE space dofs to a vector on the reduced/trace true FE space dofs.

Definition at line 388 of file staticcond.cpp.

bool mfem::StaticCondensation::ReducesTrueVSize ( ) const

Return true if applying the static condensation actually reduces the (global) number of true vector dofs.

Definition at line 118 of file staticcond.cpp.

The documentation for this class was generated from the following files:

Public Member Functions

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation