Classes and functions for multigrid preconditioners and solvers. More...

Classes
class	Kaskade::AdditiveMultiGrid< Smoother, Prolongation, CoarsePreconditioner >
	An additive multigrid preconditioner for P1 elements. More...

class	Kaskade::BDDC::InterfaceAverages< m, Index >
	A class for computing and providing BDDC coarse space constraints (interface averages). More...

class	Kaskade::BDDC::Subdomain< m, Scalar_, CoarseScalar, Transfer >
	A class representing small KKT systems on a single subdomain. More...

class	Kaskade::BDDC::SharedMemoryDomain< Subdomain >
	A class that orchestrates the exchange of data between BDDC subdomains. More...

class	Kaskade::BDDC::KKTSolver< Domain >
	A class for solving a block-diagonal-structured KKT system. More...

class	Kaskade::BDDC::BDDCSolver< Subdomain >
	Balanced Domain Decomposition with Constraints preconditioner. More...

class	Kaskade::BDDC::SpaceTransferBase< m, Scalar >
	Base class for BDDC SpaceTransfer implementations. More...

class	Kaskade::BDDC::SpaceTransfer< m, Scalar, TransmissionScalar >
	A trivial BDDC space transfer. More...

class	Kaskade::MultiplicativeMultiGrid< Entry, Index, Smoother, Prolongation >
	A general multiplicative multigrid preconditioner. More...

class	Kaskade::MGProlongation
	A prolongation operator for P1 finite elements from a coarser grid level to the next finer level. More...

class	Kaskade::MultiGridStack< Prolongation, Entry, Index >
	Class for multigrid stacks. More...

Typedefs
using	Kaskade::BDDC::Interfaces = std::vector< std::pair< int, std::vector< int > > >
	A data structure describing the interfaces of a subdomain. More...

Enumerations
enum	Kaskade::BDDC::InterfaceType { Kaskade::BDDC::CORNER = 1 , Kaskade::BDDC::EDGE = 2 , Kaskade::BDDC::FACE = 4 }
	Discrimination of different interface types. More...

Functions
template<class Index >
std::vector< std::vector< Index > >	Kaskade::BDDC::algebraicDomainDecomposition (NumaCRSPattern< Index > const &A, int n)
	Creates a decomposition of indices into overlapping subsets that can be used for constructing nonoverlapping domain decomposition solvers. More...

template<class Scalar , int m, class Index >
std::tuple< std::vector< NumaBCRSMatrix< Dune::FieldMatrix< Scalar, m, m >, Index > >, std::vector< Dune::BlockVector< Dune::FieldVector< Scalar, m > > >, std::vector< std::vector< Index > >, std::vector< std::vector< LocalDof > > >	Kaskade::BDDC::matrixDomainDecommposition (NumaBCRSMatrix< Dune::FieldMatrix< Scalar, m, m >, Index > const &A, Dune::BlockVector< Dune::FieldVector< Scalar, m > > const &f, int n, Scalar tolerance=1000 *std::numeric_limits< Scalar >::epsilon(), int maxIter=100)

template<class Entry , class Index , class Prolongation , class MakeSmoother , class MakeCoarsePreconditioner >
auto	Kaskade::makeAdditiveMultiGrid (NumaBCRSMatrix< Entry, Index > const &A, std::vector< Prolongation > const &prolongations, MakeSmoother const &makeSmoother, MakeCoarsePreconditioner const &makeCoarsePreconditioner, bool onlyLowerTriangle=false)
	Convenience function creating additive multigrid preconditioners for P1 elements. More...

template<class Entry , class Index , class GridMan >
auto	Kaskade::makeBPX (NumaBCRSMatrix< Entry, Index > const &A, GridMan const &gridman, bool onlyLowerTriangle=false)
	Convenience function creating a BPX preconditioner for P1 elements. More...

template<class Entry , class Index , class FineSpace , class MakeSmoother , class MakeCoarseSmoother >
auto	Kaskade::makeAdditivePMultiGrid (NumaBCRSMatrix< Entry, Index > A, FineSpace const &space, H1Space< typename FineSpace::Grid > const &p1Space, MakeSmoother const &makeSmoother, MakeCoarseSmoother const &makeCoarseSmoother, int coarseLevel=0, bool onlyLowerTriangle=false)
	Convenience function creating additive multigrid preconditioner of V-cycle type for higher order elements. More...

template<class StorageTag , class Entry , class Index , class FineSpace >
auto	Kaskade::makePBPX (NumaBCRSMatrix< Entry, Index > A, FineSpace const &space, H1Space< typename FineSpace::Grid > const &p1Space, StorageTag storageTag, int coarseLevel=0, bool onlyLowerTriangle=false)
	Convenience function creating additive multigrid preconditioner of V-cycle type with domain decomposition and Jacobi smoother for higher order elements. More...

template<class Entry , class Index , class Space >
auto	Kaskade::makeMultigrid (NumaBCRSMatrix< Entry, Index > const &A, Space const &space, bool onlyLowerTriangle=false, DirectType directType=DirectType::MUMPS)
	Convenience function for creating a (multiplicative) multigrid preconditioner for elliptic equations. More...

template<typename Matrix >
auto	Kaskade::makeDirectPreconditioner (Matrix &&A, DirectType directType=DirectType::MUMPS)
	Creates a direct solver for the given matrix. More...

template<int domainBlockSize, int rangeBlockSize, class Index >
auto	Kaskade::makeDirectPreconditioner (NumaBCRSMatrix< Dune::FieldMatrix< float, rangeBlockSize, domainBlockSize >, Index > &&A, DirectType directType=DirectType::MUMPS)
	Creates a direct solver for the given matrix. This specialization is for matrices containing floats (single precision) as scalar entries. Since the Kaskade interface for direct solvers only works for doubles, the matrix is copied into a new matrix with double as scalar entries. Since the coarse grid matrix often is much smaller, this hopefully means neglectable overhead. More...

template<class SparseIndex = size_t, class CoarseSpace , class FineSpace >
NumaBCRSMatrix< Dune::FieldMatrix< typename FineSpace::Scalar, 1, 1 >, SparseIndex >	Kaskade::prolongation (CoarseSpace const &coarseSpace, FineSpace const &fineSpace)
	Computes an interpolation-based prolongation matrix from a (supposedly) coarser space to a finer space. More...

template<class SparseIndex , class Mapper >
std::vector< NumaBCRSMatrix< Dune::FieldMatrix< typename Mapper::Scalar, 1, 1 >, SparseIndex > >	Kaskade::prolongationStack (FEFunctionSpace< Mapper > const &space)
	Computes a stack of prolongation matrices for higher order finite element spaces. More...

template<class Entry , class Index >
auto	Kaskade::conjugation (MGProlongation const &p, NumaBCRSMatrix< Entry, Index > const &a, bool onlyLowerTriangle=false)
	Creates a Galerkin projected Matrix \( P^T A P \) from a prolongation \( P \) and a symmetric matrix \( A \). More...

template<class GridMan >
std::vector< MGProlongation >	Kaskade::prolongationStack (GridMan const &gridman, int coarseLevel=0, size_t minNodes=0)
	Computes a sequence of prolongation matrices for P1 finite elements in hierarchical grids. More...

template<class GridMan >
std::vector< NumaBCRSMatrix< Dune::FieldMatrix< double, 1, 1 > > >	Kaskade::deformedProlongationStack (GridMan const &gridman, int coarseLevel=0, size_t minNodes=0, bool interpolateAtShiftedPosition=true)
	Computes a sequence of prolongation matrices for P1 finite elements in hierarchical grids. More...

template<class GridMan , class Entry , class Index >
MultiGridStack< MGProlongation, Entry, Index >	Kaskade::makeGeometricMultiGridStack (GridMan const &gridManager, NumaBCRSMatrix< Entry, Index > &&A, size_t minNodes=10000, bool onlyLowerTriangle=false)
	convenience routine for creating multigrid stacks based on geometric coarsening for P1 elements More...

template<class Entry , class Index >
MultiGridStack< MGProlongation, Entry, Index >	Kaskade::makeAlgebraicMultigridStack (NumaBCRSMatrix< Entry, Index > &&A, Index n=0, bool onlyLowerTriangle=false)
	Creates stack of prolongations and projected Galerkin matrices. More...

template<typename FineSpace >
auto	Kaskade::makeDeepProlongationStack (FineSpace const &space, int coarseLevel=0, int minNodes=0)
	Convenience routine for creating a deep prolongation stack from coarse to fine grid and on from P1 to higher order. More...

template<typename FineSpace , typename Matrix >
auto	Kaskade::makePMultiGridStack (FineSpace const &space, Matrix &&A, bool onlyLowerTriangle)
	convenience routine for creating multigrid stacks based on coarsening by reducing the ansatz order from P to P1 More...

template<typename FineSpace , typename CoarseSpace , typename Matrix >
auto	Kaskade::makePMultiGridStack (FineSpace const &fineSpace, Matrix &&fA, CoarseSpace const &coarseSpace)
	convenience routine for creating multigrid stacks between two spaces. More...

template<typename FineSpace , typename CoarseSpace , typename Matrix >
auto	Kaskade::makePMultiGridStack (FineSpace const &fineSpace, Matrix &&fA, CoarseSpace const &coarseSpace, Matrix &&cA)
	convenience routine for creating multigrid stacks between two spaces. More...

template<class Entry , class Index , class Prolongation , class MakeSmoother , class CoarsePreconditioner >
auto	makeMultiplicativeMultiGrid (MultiGridStack< Prolongation, Entry, Index > &&mgStack, MakeSmoother const &makeSmoother, std::unique_ptr< CoarsePreconditioner > &&coarsePreconditioner, int nPre=3, int nPost=3)
	Convenience function creating multiplicative multigrid preconditioner of V-cycle type for P1 elements. More...

template<class Entry , class Index , class GridMan >
auto	makeJacobiMultiGrid (NumaBCRSMatrix< Entry, Index > const &A, GridMan const &gridman, int nPre=3, int nPost=3, bool onlyLowerTriangle=false, DirectType directType=DirectType::MUMPS)
	Convenience function creating multiplicative Jacobi multigrid for P1 elements. More...

template<class Entry , class Index , class Space , class MakeSmoother >
auto	makeMultiplicativePMultiGrid (NumaBCRSMatrix< Entry, Index > &&A, Space const &space, MakeSmoother const &makeSmoother, int nPre=3, int nPost=3, bool onlyLowerTriangle=false, DirectType directType=DirectType::MUMPS)
	Convenience function creating multiplicative multigrid preconditioner of V-cycle type for higher order elements. More...

template<class Entry , class Index , class Space , class CoarseSpace >
auto	makeBlockJacobiPMultiGrid (NumaBCRSMatrix< Entry, Index > A, Space const &space, NumaBCRSMatrix< Entry, Index > coarseA, CoarseSpace const &coarseSpace, int nPre=3, int nPost=3, DirectType directType=DirectType::MUMPS)
	Convenience function creating multiplicative multigrid preconditioner of V-cycle type for higher order elements. More...

template<class Entry , class Index , class Space >
auto	makeBlockJacobiPMultiGrid (NumaBCRSMatrix< Entry, Index > A, Space const &space, int nPre=3, int nPost=3, bool onlyLowerTriangle=false, DirectType directType=DirectType::MUMPS)
	Convenience function creating multiplicative multigrid preconditioner of V-cycle type for higher order elements. More...

template<class Entry , class Index , class FineSpace >
auto	makeJacobiPMultiGrid (NumaBCRSMatrix< Entry, Index > A, FineSpace const &space, int nPre=3, int nPost=3, bool onlyLowerTriangle=false, DirectType directType=DirectType::MUMPS)
	Convenience function creating multiplicative multigrid preconditioner of V-cycle type with Jacobi smoother for higher order elements. More...

template<class Prolongation , class Entry , class Index >
MultiGridStack< Prolongation, Entry, Index >	makeMultiGridStack (std::vector< Prolongation > &&ps, NumaBCRSMatrix< Entry, Index > &&A, bool onlyLowerTriangle)
	Convenience routine for creating multigrid stacks. More...

Detailed Description

Classes and functions for multigrid preconditioners and solvers.

This includes general multiplicative V-cycles and additive multigrid methods, as well as several convenience functions for defining concrete preconditioners by specifying particular ingredients (smoothers and coarse grid preconditioners, in particular). Multigrid methods for arbitrary finite element order on simplicial grids are available.

If unsure which multigrid method to create, try the default construction makeMultigrid, which tries to come up with reasonable ingredients.

Typedef Documentation

◆ Interfaces

using Kaskade::BDDC::Interfaces = typedef std::vector<std::pair<int,std::vector<int> >>

A data structure describing the interfaces of a subdomain.

It consists of a sequence of pairs \( (i,idx) \) of adjacent subdomain number \( i \) and the (local to the current subdomain) degrees of freedom (sorted in increasing order) that are shared with subdomain \( i \). The sequence shall be sorted by increasing subdomain number.

Definition at line 34 of file bddcSpaceTransfer.hh.

Enumeration Type Documentation

◆ InterfaceType

enum Kaskade::BDDC::InterfaceType

Discrimination of different interface types.

Enumerator
CORNER
EDGE
FACE

Definition at line 64 of file bddc.hh.

Function Documentation

◆ algebraicDomainDecomposition()

template<class Index >

std::vector< std::vector< Index > > Kaskade::BDDC::algebraicDomainDecomposition	(	NumaCRSPattern< Index > const &	A,
		int	n
	)

Creates a decomposition of indices into overlapping subsets that can be used for constructing nonoverlapping domain decomposition solvers.

Template Parameters

Index the integral number type to be used for indices

Parameters

A	the symmetric sparse matrix pattern
n	the number of subdomains to create

Returns: a sequence of subdomains (index subsets)

Referenced by Kaskade::BDDC::matrixDomainDecommposition().

◆ conjugation()

template<class Entry , class Index >

auto Kaskade::conjugation	(	MGProlongation const &	p,
		NumaBCRSMatrix< Entry, Index > const &	a,
		bool	onlyLowerTriangle = `false`
	)

Creates a Galerkin projected Matrix \( P^T A P \) from a prolongation \( P \) and a symmetric matrix \( A \).

Parameters

onlyLowerTriangle if true, A contains onl the lower triangular part of the symmetric matrix.

Definition at line 421 of file prolongation.hh.

◆ deformedProlongationStack()

template<class GridMan >

std::vector< NumaBCRSMatrix< Dune::FieldMatrix< double, 1, 1 > > > Kaskade::deformedProlongationStack	(	GridMan const &	gridman,
		int	coarseLevel = `0`,
		size_t	minNodes = `0`,
		bool	interpolateAtShiftedPosition = `true`
	)

Computes a sequence of prolongation matrices for P1 finite elements in hierarchical grids.

We assume that each vertex not contained in the coarse grid has been created by bisecting an edge, and possibly shifted a moderate distance.

If a vertex is shifted, interpolating their direct parent nodes leads to nonlinear interpolation, i.e. coarse grid functions which are not piecwise linear on the (deformed) coarse grid cells. This can lead to high energy of the coarse grid cells. We therefore interpolate the vertex value not only from its direct parents, but from all corners of a father cell.

Template Parameters

Grid	the Dune grid type on which the P1 space is defined. The grid has to be a simplicial grid.

Parameters

grid	the grid itself
coarseLevel	the grid level to use as coarsest level
minNodes	minimum number of nodes to keep as coarse grid
interpolateAtShiftedPosition	whether or not potential vertex shifts shall be respected or not

Definition at line 644 of file prolongation.hh.

Referenced by Kaskade::makeDeepProlongationStack().

◆ makeAdditiveMultiGrid()

template<class Entry , class Index , class Prolongation , class MakeSmoother , class MakeCoarsePreconditioner >

auto Kaskade::makeAdditiveMultiGrid	(	NumaBCRSMatrix< Entry, Index > const &	A,
		std::vector< Prolongation > const &	prolongations,
		MakeSmoother const &	makeSmoother,
		MakeCoarsePreconditioner const &	makeCoarsePreconditioner,
		bool	onlyLowerTriangle = `false`
	)

Convenience function creating additive multigrid preconditioners for P1 elements.

Parameters

A	the sparse Galerkin matrix for P1 finite elements on the leaf view of the given simplicial grid
grid	a simplicial grid
makeSmoother	a callable object taking a (projected) Galerkin matrix of type NumaBCRSMatrix<Entry,Index> and a level, returning a preconditioner
onlyLowerTriangle	if true, A is assumed to be symmetric and only its lower triangular part is accessed

See also: AdditiveMultiGrid

Definition at line 159 of file additiveMultigrid.hh.

Referenced by Kaskade::makeAdditivePMultiGrid(), and Kaskade::makeBPX().

◆ makeAdditivePMultiGrid()

template<class Entry , class Index , class FineSpace , class MakeSmoother , class MakeCoarseSmoother >

auto Kaskade::makeAdditivePMultiGrid	(	NumaBCRSMatrix< Entry, Index >	A,
		FineSpace const &	space,
		H1Space< typename FineSpace::Grid > const &	p1Space,
		MakeSmoother const &	makeSmoother,
		MakeCoarseSmoother const &	makeCoarseSmoother,
		int	coarseLevel = `0`,
		bool	onlyLowerTriangle = `false`
	)

Convenience function creating additive multigrid preconditioner of V-cycle type for higher order elements.

This is realized as an additive h-multigrid nested within a two-grid scheme. The outer two-level algorithm uses the smoother obtained from

makeSmoother on the p-elements defined by the
space, and uses the h-multigrid as a coarse level preconditioner/solver.

Note that a pure Jacobi preconditioner becomes quickly inefficient when the polynomial ansatz order increases. From degree 3 on, using an overlapping domain decomposition smoother such as PatchDomainDecompositionPreconditioner is probably more efficient.

Parameters

A	the sparse Galerkin matrix for higher order finite elements on the leaf view of the given simplicial grid
space	a higher order finite element space
p1Space	a linear finite element space
makeSmoother	a callable object taking a (projected) Galerkin matrix of type NumaBCRSMatrix<Entry,Index> and a level, returning a preconditioner
makeCoarseSmoother	a callable object taking a (projected) Galerkin matrix of type NumaBCRSMatrix<Entry,Index> and a level, returning a preconditioner
coarseLevel	the grid level down to which the additive h-multigrid shall reach
onlyLowerTriangle	if true, A is assumed to be symmetric and only its lower triangular part is accessed

See also: AdditivePMultiGrid; PatchDomainDecompositionPreconditioner

Definition at line 216 of file additiveMultigrid.hh.

Referenced by Kaskade::makePBPX().

◆ makeAlgebraicMultigridStack()

template<class Entry , class Index >

MultiGridStack< MGProlongation, Entry, Index > Kaskade::makeAlgebraicMultigridStack	(	NumaBCRSMatrix< Entry, Index > &&	A,
		Index	n = `0`,
		bool	onlyLowerTriangle = `false`
	)

Creates stack of prolongations and projected Galerkin matrices.

Parameters

A	the symmetric sparse matrix
n	stop the coarsening if the number of rows/cols drops below this number
onlyLowerTriangle	if true, only the lower triangular part of symmetric A is accessed

The prolongation/restriction used is particularly simple: Stepping through the nodes, every not yet classified node is classified as coarse node, and its direct neighbours as fine nodes. Thus every fine node has at least one coarse neighbour, and the coarse nodes are not adjacent.

For each fine node, the at most two neighbouring coarse nodes with largest connection strength are selected as parents. From these, the fine node is interpolated.

Todo:: Currently, the interpolation is just with factor 0.5 from both coarse parents - not really clever for unstructured grids or jumping coefficients. Implement a more clever weighting.

◆ makeBlockJacobiPMultiGrid() [1/2]

template<class Entry , class Index , class Space >

auto makeBlockJacobiPMultiGrid	(	NumaBCRSMatrix< Entry, Index >	A,
		Space const &	space,
		int	nPre = `3`,
		int	nPost = `3`,
		bool	onlyLowerTriangle = `false`,
		DirectType	directType = `DirectType::MUMPS`
	)

This is realized as a multiplicative h-multigrid nested within a two-grid scheme. The outer two-level algorithm uses an overlapping domain decomposition smoother on the patches around grid vertices. Classical h-multigrid with point Jacobi smoother is used as a coarse level preconditioner/solver.

The default smoother step sizes are 0.5 for the domain decomposition smoother and 0.8 for the Jacobi smoother. In a preliminary test (Poisson equation on the unit square, 2016-01) these values turned out to be quite good. But for 3D 0.4 seems to be better in elastomechanics with flat geometry.

Parameters

A	the sparse Galerkin matrix for higher order finite elements on the leaf view of the given simplicial grid
space	a higher order finite element space
nPre	the number of pre-smoothings
nPost	the number of post-smoothings
onlyLowerTriangle	if true, A is assumed to be symmetric and only its lower triangular part is accessed

Definition at line 581 of file multiplicativeMultigrid.hh.

◆ makeBlockJacobiPMultiGrid() [2/2]

template<class Entry , class Index , class Space , class CoarseSpace >

auto makeBlockJacobiPMultiGrid	(	NumaBCRSMatrix< Entry, Index >	A,
		Space const &	space,
		NumaBCRSMatrix< Entry, Index >	coarseA,
		CoarseSpace const &	coarseSpace,
		int	nPre = `3`,
		int	nPost = `3`,
		DirectType	directType = `DirectType::MUMPS`
	)

This is realized as a multiplicative h-multigrid nested within a two-grid scheme. The outer two-level algorithm uses an overlapping domain decomposition smoother on the patches around grid vertices. Classical h-multigrid with point Jacobi smoother is used as a coarse level preconditioner/solver.

The default smoother step sizes are 0.5 for the domain decomposition smoother and 0.8 for the Jacobi smoother. In a preliminary test (Poisson equation on the unit square, 2016-01) these values turned out to be quite good. But for 3D 0.4 seems to be better in elastomechanics with flat geometry.

Parameters

A	the sparse Galerkin matrix for higher order finite elements on the leaf view of the given simplicial grid
space	a higher order finite element space
nPre	the number of pre-smoothings
nPost	the number of post-smoothings
onlyLowerTriangle	if true, A is assumed to be symmetric and only its lower triangular part is accessed

Definition at line 537 of file multiplicativeMultigrid.hh.

◆ makeBPX()

template<class Entry , class Index , class GridMan >

auto Kaskade::makeBPX	(	NumaBCRSMatrix< Entry, Index > const &	A,
		GridMan const &	gridman,
		bool	onlyLowerTriangle = `false`
	)

Convenience function creating a BPX preconditioner for P1 elements.

Parameters

A	the sparse Galerkin matrix for P1 finite elements on the leaf view of the given simplicial grid
grid	a simplicial grid
onlyLowerTriangle	if true, A is assumed to be symmetric and only its lower triangular part is accessed

See also: AdditiveMultiGrid

Definition at line 180 of file additiveMultigrid.hh.

◆ makeDeepProlongationStack()

template<typename FineSpace >

auto Kaskade::makeDeepProlongationStack	(	FineSpace const &	space,
		int	coarseLevel = `0`,
		int	minNodes = `0`
	)

Convenience routine for creating a deep prolongation stack from coarse to fine grid and on from P1 to higher order.

This relies on a geometric grid hierarchy for the P1 substack.

The multigrid recursion towards coarser levels is terminated as soon as the given coarse level is reached or if the given number of nodes is no longer reached.

Definition at line 921 of file prolongation.hh.

Referenced by Kaskade::makeJointPstack().

◆ makeDirectPreconditioner() [1/2]

template<typename Matrix >

auto Kaskade::makeDirectPreconditioner	(	Matrix &&	A,
		DirectType	directType = `DirectType::MUMPS`
	)

Creates a direct solver for the given matrix.

Definition at line 383 of file multiplicativeMultigrid.hh.

Referenced by Kaskade::MultiplicativeMultiGrid< Entry, Index, Smoother, Prolongation >::makeJacobiMultiGrid().

◆ makeDirectPreconditioner() [2/2]

template<int domainBlockSize, int rangeBlockSize, class Index >

auto Kaskade::makeDirectPreconditioner	(	NumaBCRSMatrix< Dune::FieldMatrix< float, rangeBlockSize, domainBlockSize >, Index > &&	A,
		DirectType	directType = `DirectType::MUMPS`
	)

Creates a direct solver for the given matrix. This specialization is for matrices containing floats (single precision) as scalar entries. Since the Kaskade interface for direct solvers only works for doubles, the matrix is copied into a new matrix with double as scalar entries. Since the coarse grid matrix often is much smaller, this hopefully means neglectable overhead.

Definition at line 436 of file multiplicativeMultigrid.hh.

◆ makeGeometricMultiGridStack()

template<class GridMan , class Entry , class Index >

MultiGridStack< MGProlongation, Entry, Index > Kaskade::makeGeometricMultiGridStack	(	GridMan const &	gridManager,
		NumaBCRSMatrix< Entry, Index > &&	A,
		size_t	minNodes = `10000`,
		bool	onlyLowerTriangle = `false`
	)

convenience routine for creating multigrid stacks based on geometric coarsening for P1 elements

Definition at line 882 of file prolongation.hh.

Referenced by Kaskade::MultiplicativeMultiGrid< Entry, Index, Smoother, Prolongation >::makeJacobiMultiGrid().

◆ makeJacobiMultiGrid()

template<class Entry , class Index , class GridMan >

auto makeJacobiMultiGrid	(	NumaBCRSMatrix< Entry, Index > const &	A,
		GridMan const &	gridman,
		int	nPre = `3`,
		int	nPost = `3`,
		bool	onlyLowerTriangle = `false`,
		DirectType	directType = `DirectType::MUMPS`
	)

This creates a V-cycle multigrid preconditioner with Jacobi smoother. A direct solver is used for grid level 0.

Parameters

A	the sparse Galerkin matrix for P1 finite elements on the leaf view of the given simplicial grid
gridman	a grid manager base of a simplicial grid
nPre	the number of pre-smoothings
nPost	the number of post-smoothings
onlyLowerTriangle	if true, A is assumed to be symmetric and only its lower triangular part is accessed

Definition at line 460 of file multiplicativeMultigrid.hh.

◆ makeJacobiPMultiGrid()

template<class Entry , class Index , class FineSpace >

auto makeJacobiPMultiGrid	(	NumaBCRSMatrix< Entry, Index >	A,
		FineSpace const &	space,
		int	nPre = `3`,
		int	nPost = `3`,
		bool	onlyLowerTriangle = `false`,
		DirectType	directType = `DirectType::MUMPS`
	)

Parameters

A	the sparse Galerkin matrix for higher order finite elements on the leaf view of the given simplicial grid
space	a higher order finite element space
nPre	the number of pre-smoothings
nPost	the number of post-smoothings
onlyLowerTriangle	if true, A is assumed to be symmetric and only its lower triangular part is accessed

Definition at line 603 of file multiplicativeMultigrid.hh.

◆ makeMultigrid()

template<class Entry , class Index , class Space >

auto Kaskade::makeMultigrid	(	NumaBCRSMatrix< Entry, Index > const &	A,
		Space const &	space,
		bool	onlyLowerTriangle = `false`,
		DirectType	directType = `DirectType::MUMPS`
	)

Convenience function for creating a (multiplicative) multigrid preconditioner for elliptic equations.

This function chooses sensible default values for the structure of the preconditioner depending in particular on the finite element ansatz order and the grid refinement depth. As the resulting preconditioners have different types depending on runtime information, we return a unique pointer to SymmetricPreconditioner. Use this if you don't have more specific ideas on how you would like the multigrid to work.

The function assumes the space consists of globally continuous finite element functions on simplicial grids with scalar shape functions.

Definition at line 40 of file multigrid.hh.

◆ makeMultiGridStack()

template<class Prolongation , class Entry , class Index >

MultiGridStack< Prolongation, Entry, Index > makeMultiGridStack	(	std::vector< Prolongation > &&	ps,
		NumaBCRSMatrix< Entry, Index > &&	A,
		bool	onlyLowerTriangle
	)

Given a stack of prolongations and the top level Galerkin matrix, this creates the complete stack including all projected Galerkin matrices.

Parameters

ps	a vector of prolongation matrices
A	the top level (finest) Galerkin matrix to be projected down
onlyLowerTriangle	if true, only the lower triangular part of A will be referenced

Definition at line 871 of file prolongation.hh.

◆ makeMultiplicativeMultiGrid()

template<class Entry , class Index , class Prolongation , class MakeSmoother , class CoarsePreconditioner >

auto makeMultiplicativeMultiGrid	(	MultiGridStack< Prolongation, Entry, Index > &&	mgStack,
		MakeSmoother const &	makeSmoother,
		std::unique_ptr< CoarsePreconditioner > &&	coarsePreconditioner,
		int	nPre = `3`,
		int	nPost = `3`
	)

Parameters

mgStack	a stack of prolongations and matching projected Galerkin matrices
makeSmoother	a callable object taking a (projected) Galerkin matrix of type NumaBCRSMatrix<Entry,Index> and a level, returning a preconditioner
coarsePreconditioner	a symmetric preconditioner to be used for "solving" on the coarsest level
nPre	the number of pre-smoothings
nPost	the number of post-smoothings

Definition at line 320 of file multiplicativeMultigrid.hh.

◆ makeMultiplicativePMultiGrid()

template<class Entry , class Index , class Space , class MakeSmoother >

auto makeMultiplicativePMultiGrid	(	NumaBCRSMatrix< Entry, Index > &&	A,
		Space const &	space,
		MakeSmoother const &	makeSmoother,
		int	nPre = `3`,
		int	nPost = `3`,
		bool	onlyLowerTriangle = `false`,
		DirectType	directType = `DirectType::MUMPS`
	)

Parameters

A	the sparse Galerkin matrix for higher order finite elements on the leaf view of the given simplicial grid
space	a higher order finite element space
p1Space	a linear finite element space
makeSmoother	a callable object taking a (projected) Galerkin matrix of type NumaBCRSMatrix<Entry,Index> and a level, returning a preconditioner
nPre	the number of pre-smoothings
nPost	the number of post-smoothings
onlyLowerTriangle	if true, A is assumed to be symmetric and only its lower triangular part is accessed

Definition at line 496 of file multiplicativeMultigrid.hh.

◆ makePBPX()

template<class StorageTag , class Entry , class Index , class FineSpace >

auto Kaskade::makePBPX	(	NumaBCRSMatrix< Entry, Index >	A,
		FineSpace const &	space,
		H1Space< typename FineSpace::Grid > const &	p1Space,
		StorageTag	storageTag,
		int	coarseLevel = `0`,
		bool	onlyLowerTriangle = `false`
	)

Convenience function creating additive multigrid preconditioner of V-cycle type with domain decomposition and Jacobi smoother for higher order elements.

Template Parameters

StorageTag a tag type for selecting the storage of inverse local matrices in the overlapping Schwarz smoother

Parameters

A	the sparse Galerkin matrix for higher order finite elements on the leaf view of the given simplicial grid
space	a higher order finite element space
p1Space	a linear finite element space
coarseLevel	the grid level down to which the additive h-multigrid shall reach
onlyLowerTriangle	if true, A is assumed to be symmetric and only its lower triangular part is accessed

The space is decomposed in (i) overlapping patches, (ii) a P1 mesh hierarchy, and (iii) a coarse space. For (i), direct solvers are used, resulting in an overlapping additive Schwarz smoother. For (ii), standard point Jacobi with damping factor 0.6 is used. (iii) is again treated with a direct solver.

See also: AdditivePMultiGrid

Definition at line 254 of file additiveMultigrid.hh.

◆ makePMultiGridStack() [1/3]

template<typename FineSpace , typename CoarseSpace , typename Matrix >

auto Kaskade::makePMultiGridStack	(	FineSpace const &	fineSpace,
		Matrix &&	fA,
		CoarseSpace const &	coarseSpace
	)

convenience routine for creating multigrid stacks between two spaces.

Definition at line 989 of file prolongation.hh.

◆ makePMultiGridStack() [2/3]

template<typename FineSpace , typename CoarseSpace , typename Matrix >

auto Kaskade::makePMultiGridStack	(	FineSpace const &	fineSpace,
		Matrix &&	fA,
		CoarseSpace const &	coarseSpace,
		Matrix &&	cA
	)

convenience routine for creating multigrid stacks between two spaces.

An approximation of the projected Galerkin matrix is provided. Often, this can be assembled much more efficiently than projected.

Definition at line 1002 of file prolongation.hh.

Referenced by Kaskade::makePMultiGridStack().

◆ makePMultiGridStack() [3/3]

template<typename FineSpace , typename Matrix >

auto Kaskade::makePMultiGridStack	(	FineSpace const &	space,
		Matrix &&	A,
		bool	onlyLowerTriangle
	)

convenience routine for creating multigrid stacks based on coarsening by reducing the ansatz order from P to P1

Definition at line 946 of file prolongation.hh.

Referenced by Kaskade::MultiplicativeMultiGrid< Entry, Index, Smoother, Prolongation >::makeBlockJacobiPMultiGrid(), and Kaskade::MultiplicativeMultiGrid< Entry, Index, Smoother, Prolongation >::makeMultiplicativePMultiGrid().

◆ matrixDomainDecommposition()

template<class Scalar , int m, class Index >

std::tuple< std::vector< NumaBCRSMatrix< Dune::FieldMatrix< Scalar, m, m >, Index > >, std::vector< Dune::BlockVector< Dune::FieldVector< Scalar, m > > >, std::vector< std::vector< Index > >, std::vector< std::vector< LocalDof > > > Kaskade::BDDC::matrixDomainDecommposition	(	NumaBCRSMatrix< Dune::FieldMatrix< Scalar, m, m >, Index > const &	A,
		Dune::BlockVector< Dune::FieldVector< Scalar, m > > const &	f,
		int	n,
		Scalar	tolerance = `1000*std::numeric_limits<Scalar>::epsilon()`,
		int	maxIter = `100`
	)

Parameters

tolerance	defines the termination criterion by asking for \[ \max_{k,i} \sum_j \frac{\|a_{ij}^k\|}{a_{ii}^k} -1 \le \mathrm{tol}. \]
maxIter	defines a termination criterion by imposing an upper bound on the iteration count.

Returns

Definition at line 67 of file add.hh.

◆ prolongation()

template<class SparseIndex = size_t, class CoarseSpace , class FineSpace >

NumaBCRSMatrix< Dune::FieldMatrix< typename FineSpace::Scalar, 1, 1 >, SparseIndex > Kaskade::prolongation	(	CoarseSpace const &	coarseSpace,
		FineSpace const &	fineSpace
	)

Computes an interpolation-based prolongation matrix from a (supposedly) coarser space to a finer space.

Template Parameters

CoarseSpace	a FEFunctionSpace type
FineSpace	a FEFunctionSpace type

Parameters

coarseSpace	the (supposedly) coarser space (domain)
fineSpace	the (supposedly) finer space (image)

Both function spaces have to be defined on the very same grid view and to be geometrically nested.

Definition at line 43 of file prolongation.hh.

Referenced by Kaskade::TaylorHoodPreconditioner< X >::apply(), Kaskade::localProlongationMatrix(), and Kaskade::TransferData< Space, CoarseningPolicy >::transferMatrix().

◆ prolongationStack() [1/2]

template<class SparseIndex , class Mapper >

std::vector< NumaBCRSMatrix< Dune::FieldMatrix< typename Mapper::Scalar, 1, 1 >, SparseIndex > > Kaskade::prolongationStack ( FEFunctionSpace< Mapper > const & space )

Computes a stack of prolongation matrices for higher order finite element spaces.

This computes a stack of prolongation matrices for a scale of finite element spaces with increasing polynomial ansatz order. From level to level, the ansatz order is doubled (maybe +1 for odd degrees).

Template Parameters

Mapper a finite element local to global mapper with scalar shape functions.

Definition at line 131 of file prolongation.hh.

Referenced by Kaskade::makeAdditivePMultiGrid(), Kaskade::makeBPX(), Kaskade::makeGeometricMultiGridStack(), and Kaskade::prolongationStack().

◆ prolongationStack() [2/2]

template<class GridMan >

std::vector< MGProlongation > Kaskade::prolongationStack	(	GridMan const &	gridman,
		int	coarseLevel = `0`,
		size_t	minNodes = `0`
	)

Computes a sequence of prolongation matrices for P1 finite elements in hierarchical grids.

We assume that each vertex not contained in the coarse grid has been created by bisecting an edge, such that there are exactly two "parent vertices".

Template Parameters

Grid	the Dune grid type on which the P1 space is defined. The grid has to be a simplicial grid.

Parameters

grid	the grid itself
minNodes	minimum number of nodes to keep as coarse grid

Definition at line 588 of file prolongation.hh.

Classes

Typedefs

Enumerations

Functions

Detailed Description

Typedef Documentation

◆ Interfaces

Enumeration Type Documentation

◆ InterfaceType

Function Documentation

◆ algebraicDomainDecomposition()

◆ conjugation()

◆ deformedProlongationStack()

◆ makeAdditiveMultiGrid()

◆ makeAdditivePMultiGrid()

◆ makeAlgebraicMultigridStack()

◆ makeBlockJacobiPMultiGrid() [1/2]

◆ makeBlockJacobiPMultiGrid() [2/2]

◆ makeBPX()

◆ makeDeepProlongationStack()

◆ makeDirectPreconditioner() [1/2]

◆ makeDirectPreconditioner() [2/2]

◆ makeGeometricMultiGridStack()

◆ makeJacobiMultiGrid()

◆ makeJacobiPMultiGrid()

◆ makeMultigrid()

◆ makeMultiGridStack()

◆ makeMultiplicativeMultiGrid()

◆ makeMultiplicativePMultiGrid()

◆ makePBPX()

◆ makePMultiGridStack() [1/3]

◆ makePMultiGridStack() [2/3]

◆ makePMultiGridStack() [3/3]

◆ matrixDomainDecommposition()

◆ prolongation()

◆ prolongationStack() [1/2]

◆ prolongationStack() [2/2]