#include <deal.II/lac/schur_matrix.h>

Inheritance diagram for SchurMatrix< MA_inverse, MB, MDt, MC >:

Public Member Functions
	SchurMatrix (const MA_inverse &Ainv, const MB &B, const MDt &Dt, const MC &C, VectorMemory< BlockVector< double > > &mem, const std::vector< types::global_dof_index > &signature=std::vector< types::global_dof_index >(0))

void	prepare_rhs (BlockVector< double > &dst, const BlockVector< double > &src) const

void	vmult (BlockVector< double > &dst, const BlockVector< double > &src) const

double	residual (BlockVector< double > &dst, const BlockVector< double > &src, const BlockVector< double > &rhs) const

void	postprocess (BlockVector< double > &dst, const BlockVector< double > &src, const BlockVector< double > &rhs) const

void	debug_level (unsigned int l)

Public Member Functions inherited from Subscriptor
	Subscriptor ()

	Subscriptor (const Subscriptor &)

	Subscriptor (Subscriptor &&)

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&)

void	subscribe (const char *identifier=0) const

void	unsubscribe (const char *identifier=0) const

unsigned int	n_subscriptions () const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Private Member Functions
	SchurMatrix (const SchurMatrix< MA_inverse, MB, MDt, MC > &)

SchurMatrix &	operator= (const SchurMatrix< MA_inverse, MB, MDt, MC > &)

Private Attributes
const SmartPointer< const MA_inverse, SchurMatrix< MA_inverse, MB, MDt, MC > >	Ainv

const SmartPointer< const MB, SchurMatrix< MA_inverse, MB, MDt, MC > >	B

const SmartPointer< const MDt, SchurMatrix< MA_inverse, MB, MDt, MC > >	Dt

const SmartPointer< const MC, SchurMatrix< MA_inverse, MB, MDt, MC > >	C

VectorMemory< BlockVector< double > > &	mem

std::vector< types::global_dof_index >	signature

unsigned int	debug

Additional Inherited Members
Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, char *arg2, std::string &arg3)

static ::ExceptionBase &	ExcNoSubscriber (char arg1, char arg2)

Detailed Description

template<class MA_inverse, class MB, class MDt, class MC>
class SchurMatrix< MA_inverse, MB, MDt, MC >

Schur complement of a block matrix.

Given a non-singular matrix A (often positive definite) and a positive semi-definite matrix C as well as matrices B and Dt of full rank, this class implements a new matrix, the Schur complement a the system of equations of the structure

* /        \  /   \     /   \
* |  A  Dt |  | u |  -  | f |
* | -B  C  |  | p |  -  | g |
* \        /  \   /     \   /
*

Multiplication with the Schur matrix S is the operation

* S p = C p + B A-inverse Dt-transpose p,
*

which is an operation within the space for p.

The data handed to the Schur matrix are as follows:

A: the inverse of A is stored, instead of A. This allows the application to use the most efficient form of inversion, iterative or direct.

B, C: these matrices are stored "as is".

Dt: the computation of the Schur complement involves the function Tvmult of the matrix Dt, not vmult! This way, it is sufficient to build only one matrix B for the symmetric Schur complement and use it twice.

All matrices involved are of arbitrary type and vectors are BlockVectors. This way, SchurMatrix can be coupled with any matrix classes providing vmult and Tvmult and can be even nested. Since SmartPointers of matrices are stored, the matrix blocks should be derived from Subscriptor.

Since the Schur complement of a matrix corresponds to a Gaussian block elimination, the right hand side of the condensed system must be preprocessed. Furthermore, the eliminated variable must be reconstructed after solving.

*   g = g + B A-inverse f
*   u = A-inverse (f - D-transpose p)
*

Applying these transformations, the solution of the system above by a SchurMatrix schur is coded as follows:

schur.prepare_rhs (g, f);
solver.solve (schur, p, g, precondition);
schur.postprocess (u, p);

See also: Block (linear algebra)

Author: Guido Kanschat, 2000, 2001, 2002

Definition at line 97 of file schur_matrix.h.

Constructor & Destructor Documentation

◆ SchurMatrix() [1/2]

template<class MA_inverse , class MB , class MDt , class MC >

SchurMatrix< MA_inverse, MB, MDt, MC >::SchurMatrix	(	const MA_inverse &	Ainv,
		const MB &	B,
		const MDt &	Dt,
		const MC &	C,
		VectorMemory< BlockVector< double > > &	mem,
		const std::vector< types::global_dof_index > &	signature = `std::vector<types::global_dof_index>(0)`
	)

Constructor. This constructor receives all the matrices needed. Furthermore, it gets a reference to a memory structure for obtaining block vectors.

Optionally, the length of the u-vector can be provided.

For the meaning of the matrices see the class documentation.

Definition at line 206 of file schur_matrix.h.

◆ SchurMatrix() [2/2]

template<class MA_inverse , class MB , class MDt , class MC >

SchurMatrix< MA_inverse, MB, MDt, MC >::SchurMatrix ( const SchurMatrix< MA_inverse, MB, MDt, MC > & )

private

No copy constructor.

Member Function Documentation

◆ prepare_rhs()

template<class MA_inverse , class MB , class MDt , class MC >

void SchurMatrix< MA_inverse, MB, MDt, MC >::prepare_rhs	(	BlockVector< double > &	dst,
		const BlockVector< double > &	src
	)		const

Do block elimination of the right hand side. Given right hand sides for both components of the block system, this function provides the right hand side for the Schur complement.

The result is stored in the first argument, which is also part of the input data. If it is necessary to conserve the data, dst must be copied before calling this function. This is reasonable, since in many cases, only the pre-processed right hand side is needed.

Definition at line 282 of file schur_matrix.h.

◆ vmult()

template<class MA_inverse , class MB , class MDt , class MC >

void SchurMatrix< MA_inverse, MB, MDt, MC >::vmult	(	BlockVector< double > &	dst,
		const BlockVector< double > &	src
	)		const

Multiplication with the Schur complement.

Definition at line 231 of file schur_matrix.h.

◆ residual()

template<class MA_inverse , class MB , class MDt , class MC >

double SchurMatrix< MA_inverse, MB, MDt, MC >::residual	(	BlockVector< double > &	dst,
		const BlockVector< double > &	src,
		const BlockVector< double > &	rhs
	)		const

Computation of the residual of the Schur complement.

Definition at line 269 of file schur_matrix.h.

◆ postprocess()

template<class MA_inverse , class MB , class MDt , class MC >

void SchurMatrix< MA_inverse, MB, MDt, MC >::postprocess	(	BlockVector< double > &	dst,
		const BlockVector< double > &	src,
		const BlockVector< double > &	rhs
	)		const

Compute the eliminated variable from the solution of the Schur complement problem.

Definition at line 311 of file schur_matrix.h.

◆ debug_level()

template<class MA_inverse , class MB , class MDt , class MC >

void SchurMatrix< MA_inverse, MB, MDt, MC >::debug_level ( unsigned int l )

Select debugging information for log-file. Debug level 1 is defined and writes the norm of every vector before and after each operation. Debug level 0 turns off debugging information.

Definition at line 223 of file schur_matrix.h.

◆ operator=()

template<class MA_inverse , class MB , class MDt , class MC >

SchurMatrix& SchurMatrix< MA_inverse, MB, MDt, MC >::operator= ( const SchurMatrix< MA_inverse, MB, MDt, MC > & )

private

No assignment.

Member Data Documentation

◆ Ainv

template<class MA_inverse , class MB , class MDt , class MC >

const SmartPointer<const MA_inverse,SchurMatrix<MA_inverse,MB,MDt,MC> > SchurMatrix< MA_inverse, MB, MDt, MC >::Ainv

private

Pointer to inverse of upper left block.

Definition at line 172 of file schur_matrix.h.

◆ B

template<class MA_inverse , class MB , class MDt , class MC >

const SmartPointer<const MB,SchurMatrix<MA_inverse,MB,MDt,MC> > SchurMatrix< MA_inverse, MB, MDt, MC >::B

private

Pointer to lower left block.

Definition at line 176 of file schur_matrix.h.

◆ Dt

template<class MA_inverse , class MB , class MDt , class MC >

const SmartPointer<const MDt,SchurMatrix<MA_inverse,MB,MDt,MC> > SchurMatrix< MA_inverse, MB, MDt, MC >::Dt

private

Pointer to transpose of upper right block.

Definition at line 180 of file schur_matrix.h.

◆ C

template<class MA_inverse , class MB , class MDt , class MC >

const SmartPointer<const MC,SchurMatrix<MA_inverse,MB,MDt,MC> > SchurMatrix< MA_inverse, MB, MDt, MC >::C

private

Pointer to lower right block.

Definition at line 184 of file schur_matrix.h.

◆ mem

template<class MA_inverse , class MB , class MDt , class MC >

VectorMemory<BlockVector<double> >& SchurMatrix< MA_inverse, MB, MDt, MC >::mem

private

Auxiliary memory for vectors.

Definition at line 188 of file schur_matrix.h.

◆ signature

template<class MA_inverse , class MB , class MDt , class MC >

std::vector<types::global_dof_index> SchurMatrix< MA_inverse, MB, MDt, MC >::signature

private

Optional signature of the u-vector.

Definition at line 193 of file schur_matrix.h.

◆ debug

template<class MA_inverse , class MB , class MDt , class MC >

unsigned int SchurMatrix< MA_inverse, MB, MDt, MC >::debug

private

Switch for debugging information.

Definition at line 198 of file schur_matrix.h.

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

deal.II/lac/schur_matrix.h

Public Member Functions

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

template<class MA_inverse, class MB, class MDt, class MC> class SchurMatrix< MA_inverse, MB, MDt, MC >

Constructor & Destructor Documentation

◆ SchurMatrix() [1/2]

◆ SchurMatrix() [2/2]

Member Function Documentation

◆ prepare_rhs()

◆ vmult()

◆ residual()

◆ postprocess()

◆ debug_level()

◆ operator=()

Member Data Documentation

◆ Ainv

◆ B

◆ Dt

◆ C

◆ mem

◆ signature

◆ debug

template<class MA_inverse, class MB, class MDt, class MC>
class SchurMatrix< MA_inverse, MB, MDt, MC >