Tools for converting finite element functions. More...

Modules
	Functionviews
	Predefined function views.

Classes
class	Kaskade::AdvectedFunctionView< Function, Velocity >
	A weak function view that defines an advected function. More...

struct	Kaskade::AdaptationCoarseningPolicy
	policy class for LocalTransfer These two structures are used for determining whether to perform a "normal" grid transfer (default, AdaptionCoarseningPolicy), or to calculate prolongation matrices for a FE function space based on a HierarchicIndexSet (see mgtools.hh). More...

struct	Kaskade::MultilevelCoarseningPolicy
	policy class for LocalTransfer These two structures are used for determining whether to perform a "normal" grid transfer (default, AdaptionCoarseningPolicy), or to calculate prolongation matrices for a FE function space based on a HierarchicIndexSet (see mgtools.hh). More...

class	Kaskade::LocalTransfer< Space, CoarseningPolicy >
	Class that stores for a cell ("this entity" or "father") a local projection matrix. More...

class	Kaskade::TransferData< Space, CoarseningPolicy >
	A class storing data collected before grid adaptation that is necessary to transfer FE functions. This includes global indices of shape functions on leaf cells and restriction matrices to coarser cells in case some leaf cells might be removed during grid adaptation. More...

struct	Kaskade::InverseVolume
	A Weighing that associates to each cell its inverse volume. More...

struct	Kaskade::Volume
	A Weighing that associates to each cell its volume. More...

struct	Kaskade::PlainAverage
	A Weighing that associates to each cell a constant weight of 1. More...

class	Kaskade::WeakFunctionViewAdaptor< Functor >
	An adaptor that turns callables (e.g., lambdas) into weak function views. More...

class	Kaskade::FunctionViewAdaptor< Space_, Functor >
	An adaptor that allows to turn lambda functions into function views. More...

Functions
template<class Space , class ShapeFunctionSet >
void	Kaskade::evaluateGlobalShapeFunctions (Space const &space, Cell< typename Space::GridView > const &cell, std::vector< LocalPosition< typename Space::GridView > > const &xi, DynamicMatrix< Dune::FieldMatrix< typename Space::Scalar, Space::sfComponents, 1 > > &sfValues, ShapeFunctionSet const &shapeFunctionSet)
	Computes the values of global shape functions at the given points (which are supposed to be local coordinates inside the cell to which the evaluator is currently moved). More...

template<class Space , class ShapeFunctionSet >
void	Kaskade::approximateGlobalValues (Space const &space, typename Space::Grid::template Codim< 0 >::Entity const &cell, DynamicMatrix< Dune::FieldMatrix< typename Space::Scalar, Space::sfComponents, 1 > > &globalValues, DynamicMatrix< Dune::FieldMatrix< typename Space::Scalar, 1, 1 > > &coeff, ShapeFunctionSet const &sfs)
	Computes global shape function coefficients such that the given set of global function values at the interpolation nodes of the shape function set are approximated by the resulting linear combinations of global shape functions. More...

template<class ChildSpace , class FatherSpace >
void	Kaskade::localProlongationMatrix (ChildSpace const &childSpace, Cell< typename ChildSpace::GridView > const &child, FatherSpace const &fatherSpace, Cell< typename FatherSpace::GridView > const &father, DynamicMatrix< Dune::FieldMatrix< typename ChildSpace::Scalar, 1, 1 > > &prolongation)
	Computes a local prolongation matrix from father cell to child cell. More...

template<typename Weighing = PlainAverage, typename FSElement , typename Function >
void	Kaskade::interpolateGlobally (FSElement &fse, Function const &fu)
	Interpolates FunctionSpaceElement to FunctionSpaceElement. More...

template<typename Weighing = PlainAverage, typename Target , typename Source >
void	Kaskade::interpolateGloballyWeak (Target &target, Source const &fu)
	Interpolates WeakFunctionViews to FunctionSpaceElement. More...

template<class FromSpace , class ToSpace >
Prolongation	Kaskade::twoGridProlongation (FromSpace const &from, ToSpace const &to)
	Computes an interpolation matrix for transfer between P1 finite element spaces on two different grids. More...

template<class FromSpace , class ToSpace >
Prolongation	Kaskade::nonNestedProlongation (FromSpace const &from, ToSpace const &to, double tolTrunc=0.1)
	Computes an interpolation matrix for transfer between P1 finite element spaces on two different (possibly non-nested) grids. More...

std::vector< Prolongation >	Kaskade::nonNestedProlognationStack (std::vector< Prolongation > ps)
	Computes a full column rank prolongation stack from a colum rank deficient prolongation stack. More...

template<class Space >
std::vector< Prolongation >	Kaskade::makeNonNestedPstack (std::initializer_list< Space > spacelist, double tolTrunc=0.1)
	Computes an interpolation matrix for transfer between P1 finite element spaces on two different (possibly non-nested) grids. More...

template<class Space1 , class Space2 >
std::vector< Prolongation >	Kaskade::makeJointPstack (Space1 space1, Space2 space2, int coarseLevel=0)
	Computes a (joint) prolongation stack for a geometry defined by two grids. More...

template<class Function , class Velocity >
auto	makeAdvectedFunctionView (Function const &f, Velocity const &v, double tau, int n=1)
	Creates a weak function view for advecting FE functions. More...

template<class Functor >
auto	makeWeakFunctionView (Functor const &g)
	A convenience function turning callables (e.g., lambda functions) into weak function views. More...

template<class Space , class Functor >
auto	makeFunctionView (Space const &space, Functor const &g)
	A convenience functor that supports the easy creation of function view adaptors. More...

Detailed Description

Tools for converting finite element functions.

Function Documentation

◆ approximateGlobalValues()

template<class Space , class ShapeFunctionSet >

void Kaskade::approximateGlobalValues	(	Space const &	space,
		typename Space::Grid::template Codim< 0 >::Entity const &	cell,
		DynamicMatrix< Dune::FieldMatrix< typename Space::Scalar, Space::sfComponents, 1 > > &	globalValues,
		DynamicMatrix< Dune::FieldMatrix< typename Space::Scalar, 1, 1 > > &	coeff,
		ShapeFunctionSet const &	sfs
	)

Computes global shape function coefficients such that the given set of global function values at the interpolation nodes of the shape function set are approximated by the resulting linear combinations of global shape functions.

This realizes the matrix product \( \Phi^{+} \Psi^{-1} \) ( \( \Psi \) is the converter, and \( \Phi \) the values of the shape functions at the interpolation nodes, see LocalToGlobalMapperConcept).

On exit, the value of globalValues is undefined.

The actual meaning of "approximate" (formally denoted by a pseudoinverse symbol here) is defined by the specific FE space. However, for any space it holds that after executing

evaluateGlobalShapeFunctions(space,cell,space.mapper().shapefunctions(cell).interpolationNodes(),sfValues);

approximateGlobalValues(space,cell,sfValues,coeff);

Kaskade::approximateGlobalValues

void approximateGlobalValues(Space const &space, typename Space::Grid::template Codim< 0 >::Entity const &cell, DynamicMatrix< Dune::FieldMatrix< typename Space::Scalar, Space::sfComponents, 1 > > &globalValues, DynamicMatrix< Dune::FieldMatrix< typename Space::Scalar, 1, 1 > > &coeff, ShapeFunctionSet const &sfs)

Computes global shape function coefficients such that the given set of global function values at the ...

Definition: fetransfer.hh:133

Kaskade::evaluateGlobalShapeFunctions

void evaluateGlobalShapeFunctions(Space const &space, Cell< typename Space::GridView > const &cell, std::vector< LocalPosition< typename Space::GridView > > const &xi, DynamicMatrix< Dune::FieldMatrix< typename Space::Scalar, Space::sfComponents, 1 > > &sfValues, ShapeFunctionSet const &shapeFunctionSet)

Computes the values of global shape functions at the given points (which are supposed to be local coo...

Definition: fetransfer.hh:69

the matrix coeff is the identity matrix.

Parameters

[in]	globalValues	global function values \( f(\xi_i) \) at interpolation nodes \( \xi_i \) of the shape function set (will be overwritten and invalidated)
[out]	coeff	coefficients \( C \) of shape function linear combination such that the global values at the interpolation points are approximated: \( f(\xi_i) = \sum_j C_{ij} \phi_j(\xi_i) \)
	sfs	the space's shape function set on the given cell (note that the cell needs not be contained in the space's grid view at all, e.g. when creating multilevel prolongations)

Definition at line 133 of file fetransfer.hh.

Referenced by Kaskade::approximateGlobalValues(), Kaskade::localProlongationMatrix(), Kaskade::LocalTransfer< Space, CoarseningPolicy >::LocalTransfer(), and Kaskade::prolongation().

◆ evaluateGlobalShapeFunctions()

template<class Space , class ShapeFunctionSet >

void Kaskade::evaluateGlobalShapeFunctions	(	Space const &	space,
		Cell< typename Space::GridView > const &	cell,
		std::vector< LocalPosition< typename Space::GridView > > const &	xi,
		DynamicMatrix< Dune::FieldMatrix< typename Space::Scalar, Space::sfComponents, 1 > > &	sfValues,
		ShapeFunctionSet const &	shapeFunctionSet
	)

Computes the values of global shape functions at the given points (which are supposed to be local coordinates inside the cell to which the evaluator is currently moved).

This realizes the matrix product \(\Psi\Phi \) ( \( \Phi \) is the matrix of shape function values at the given points \( \xi \) in the reference element and \(\Psi\) is the converter, see LocalToGlobalMapperConcept). On return,

sfValues[i][j] contains the value of global shape function j at global(xi[i]).

◆ interpolateGlobally()

template<typename Weighing = PlainAverage, typename FSElement , typename Function >

void Kaskade::interpolateGlobally	(	FSElement &	fse,
		Function const &	fu
	)

Interpolates FunctionSpaceElement to FunctionSpaceElement.

This is particulary useful together with FunctionViews and makes it possible to transform a FunctionView to a FunctionSpaceElement

Parameters

fse	function space element which is set to (approximately) the value of fu
fu	function space element of a (probably different) FE space

Template Parameters

Weighing	has to be default constructible
FSElement	type of the target finite element function
Function	type of the source function

See also: makeFunctionView

Definition at line 855 of file fetransfer.hh.

Referenced by Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator=(), and Kaskade::spaceTransfer().

◆ interpolateGloballyWeak()

template<typename Weighing = PlainAverage, typename Target , typename Source >

void Kaskade::interpolateGloballyWeak	(	Target &	target,
		Source const &	fu
	)

Interpolates WeakFunctionViews to FunctionSpaceElement.

This is particulary useful together with WeakFunctionViews and makes it possible to transform a WeakFunctionView to a FunctionSpaceElement.

Template Parameters

Weighing	the type of averaging
Target	the target function type
Source	the source weak function view type

Remember that WeakFunctionView s support the evaluation by cell and local coordinate. If the source function is a finite element function, prefer interpolateGlobally, since this is a little bit faster.

Definition at line 946 of file fetransfer.hh.

Referenced by Kaskade::DeformingGridManager< Grid >::correctingDisplacement(), and Kaskade::FunctionSpaceElement< FunctionSpace, m >::operator=().

◆ localProlongationMatrix()

template<class ChildSpace , class FatherSpace >

void Kaskade::localProlongationMatrix	(	ChildSpace const &	childSpace,
		Cell< typename ChildSpace::GridView > const &	child,
		FatherSpace const &	fatherSpace,
		Cell< typename FatherSpace::GridView > const &	father,
		DynamicMatrix< Dune::FieldMatrix< typename ChildSpace::Scalar, 1, 1 > > &	prolongation
	)

Computes a local prolongation matrix from father cell to child cell.

Parameters

child	a pointer to a cell contained in the child space's grid view
father	a pointer to a cell contained in the father space's grid view

Definition at line 232 of file fetransfer.hh.

◆ makeAdvectedFunctionView()

template<class Function , class Velocity >

auto makeAdvectedFunctionView	(	Function const &	f,
		Velocity const &	v,
		double	tau,
		int	n = `1`
	)

Definition at line 214 of file advect.hh.

◆ makeFunctionView()

template<class Space , class Functor >

auto makeFunctionView	(	Space const &	space,
		Functor const &	g
	)

Template Parameters

Space	a FEFunctionSpace type
Functor	a callable object with operator()(Space::Evaluator eval)

Definition at line 1107 of file fetransfer.hh.

◆ makeJointPstack()

template<class Space1 , class Space2 >

std::vector< Prolongation > Kaskade::makeJointPstack	(	Space1	space1,
		Space2	space2,
		int	coarseLevel = `0`
	)

Computes a (joint) prolongation stack for a geometry defined by two grids.

Template Parameters

Space1	FEFunctionSpace built on gridManager of first geometry
Space2	FEFunctionSpace built on gridManager of second geometry

Parameters

space1	space of first geometry
space2	space of second geometry

This function computes a prolongation stack for a hierarchy based on two (hierarchical) spaces. At least one of the spaces should be hierarchical, i.e. have more than one level. If the number of levels of the two spaces differ, the hierarchy of the space with fewer levels is "augmeted" by identity prolongation matrices. For each of the spaces this function calls makeDeepProlongationStack(...) and concatenates the two prolongation stacks.

Definition at line 334 of file amg.hh.

◆ makeNonNestedPstack()

template<class Space >

std::vector< Prolongation > Kaskade::makeNonNestedPstack	(	std::initializer_list< Space >	spacelist,
		double	tolTrunc = `0.1`
	)

Computes an interpolation matrix for transfer between P1 finite element spaces on two different (possibly non-nested) grids.

Template Parameters

FromSpace	a FEFunctionSpace of continuous piecewise linear elements
ToSpace	a FEFunctionSpace of continuous piecewise linear elements

Parameters

from	the coarse space
to	the fine space

The computed interpolation matrix transfers coefficient vectors of the coarse space to coefficient vectors of the fine space such that the function is (approximately) the same. Thus, it is a representation of (approximate) identity w.r.t. the Lagrangian bases of the involved spaces. Since the spaces might be non-nested, there are two undesider case that can come up:

A fine grid vertex is not contained in the corase cell. To deal we this we express the fine grid DOF wrt to the DOFs "closest" to the corresponsing vertex.
In the case a coarse grid cell/DOF is not contained in the fine cell, this introduces a zero column in the prolongation matrix. Use the nonNestedProlognationStack() function to remove zero columns from a resulting prolognation stack.

Definition at line 300 of file amg.hh.

◆ makeWeakFunctionView()

template<class Functor >

auto makeWeakFunctionView ( Functor const & g )

This is a model of the WeakFunctionView concept.

Template Parameters

Functor a callable object with operator()(Cell, LocalPosition)

Definition at line 1059 of file fetransfer.hh.

◆ nonNestedProlognationStack()

std::vector< Prolongation > Kaskade::nonNestedProlognationStack ( std::vector< Prolongation > ps )

Computes a full column rank prolongation stack from a colum rank deficient prolongation stack.

Parameters

ps	vector containing the prolognation stack.

The prolognation stack obtained from non-nested meshes may contain zero columns (in particular if a coarse grid cell is not contained in the fine mesh). To avoid this, we "erase" the corresponding DOFs on an algebraic level, i.e. by deleting the corresponding column in the prolognation matrix. In the prolognation matrix on the next coarser level in the stack we need to delete the corresponding rows, too. Doing this on each of the levels we obtain a full column rank prolognation stack.

Definition at line 225 of file amg.hh.

Referenced by Kaskade::makeNonNestedPstack().

◆ nonNestedProlongation()

template<class FromSpace , class ToSpace >

Prolongation Kaskade::nonNestedProlongation	(	FromSpace const &	from,
		ToSpace const &	to,
		double	tolTrunc = `0.1`
	)

Computes an interpolation matrix for transfer between P1 finite element spaces on two different (possibly non-nested) grids.

Template Parameters

FromSpace	a FEFunctionSpace of continuous piecewise linear elements
ToSpace	a FEFunctionSpace of continuous piecewise linear elements

Parameters

from	the coarse space
to	the fine space
tolTrunc	tolerance for truncation of P-matrix entries (entries below tol*rowtotal will be deleted)

The computed interpolation matrix transfers coefficient vectors of the coarse space to coefficient vectors of the fine space such that the function is (approximately) the same. Thus, it is a representation of (approximate) identity w.r.t. the Lagrangian bases of the involved spaces. Since the spaces might be non-nested, there are two undesired cases that can come up:

A fine grid vertex is not contained in any coarse cell: To deal with this we express the fine grid DOF wrt to the DOFs "closest" to the corresponsing vertex.
A coarse grid cell/DOF is not contained any fine cell: This introduces a zero column in the prolongation matrix, which may lead to undefinite Galerkin projections. Use the nonNestedProlognationStack() function to remove zero columns from a resulting prolognation matrix or a prolongation stack.

To retain good convergence of MG methods based on this prolongation matix, a truncation of "small" entries is required. In each row, we delete all entries, which are smaller than tolTrunc*rowtotal, with rowtotal being the sum of the absolute values of the row entries. Afterwards we scale each row to retain its original row total. The value of 0.2 seems to be a good guess, which balances the two competing goals of operator complexity and a good convergence rate.

Cf. Dickopf, Krause - "Numerical Study of the Almost Nested Case in a Multlevel Method Based on Non-nested Meshes" (2013)

Definition at line 125 of file amg.hh.

Referenced by Kaskade::makeNonNestedPstack().

◆ twoGridProlongation()

template<class FromSpace , class ToSpace >

Prolongation Kaskade::twoGridProlongation	(	FromSpace const &	from,
		ToSpace const &	to
	)

Computes an interpolation matrix for transfer between P1 finite element spaces on two different grids.

Template Parameters

FromSpace	a FEFunctionSpace of continuous piecewise linear elements
ToSpace	a FEFunctionSpace of continuous piecewise linear elements

Parameters

from	the coarse space
to	the fine space

The computed interpolation matrix transfers coefficient vectors of the coarse space to coefficient vectors of the fine space such that the function is (approximately) the same. Thus, it is a representation of (approximate) identity w.r.t. the Lagrangian bases of the involved spaces.

An example use is in auxiliary space methods, where

Todo:: generalize to arbitrary spaces, not only P1?

Definition at line 51 of file amg.hh.

Referenced by Kaskade::makeAuxiliarySpaceMultigridStack().

Modules

Classes

Functions

Detailed Description

Function Documentation

◆ approximateGlobalValues()

◆ evaluateGlobalShapeFunctions()

◆ interpolateGlobally()

◆ interpolateGloballyWeak()

◆ localProlongationMatrix()

◆ makeAdvectedFunctionView()

◆ makeFunctionView()

◆ makeJointPstack()

◆ makeNonNestedPstack()

◆ makeWeakFunctionView()

◆ nonNestedProlognationStack()

◆ nonNestedProlongation()

◆ twoGridProlongation()