Function space basis implementations

Implementations of the GlobalBasis concept. More...

Classes
class	Dune::Functions::BSplineLocalBasis< GV, R >
	LocalBasis class in the sense of dune-localfunctions, presenting the restriction of a B-spline patch to a knot span. More...
class	Dune::Functions::BSplineLocalCoefficients< dim >
	Attaches a shape function to an entity. More...
class	Dune::Functions::BSplineLocalInterpolation< dim, LB >
	Local interpolation in the sense of dune-localfunctions, for the B-spline basis on tensor-product grids. More...
class	Dune::Functions::BSplineLocalFiniteElement< GV, R >
	LocalFiniteElement in the sense of dune-localfunctions, for the B-spline basis on tensor-product grids. More...
class	Dune::Functions::BSplinePreBasis< GV >
	Pre-basis for B-spline basis. More...
class	Dune::Functions::CompositePreBasis< IMS, SPB >
	A pre-basis for composite bases. More...
class	Dune::Functions::CubicHermitePreBasis< GV, reduced >
	A pre-basis for a Hermitebasis. More...
class	Dune::Functions::HierarchicalLagrangePreBasis< GV, k, R >
	A pre-basis for a hierarchical Lagrange basis. More...
class	Dune::Functions::LagrangePreBasis< GV, k, R >
	A pre-basis for a PQ-lagrange bases with given order. More...
class	Dune::Functions::LFEPreBasisMixin< GV, LFE >
	A pre-basis mixin class parametrized with a local finite-element and a DOF layout. More...
class	Dune::Functions::MorleyPreBasis< GV, R >
	A pre-basis for a Morleybasis. More...
class	Dune::Functions::RannacherTurekPreBasis< GV >
	Pre-basis for a Rannacher-Turek basis. More...
class	Dune::Functions::RefinedLagrangePreBasis< GV, k, R >
	A pre-basis for a refined Lagrange bases. More...
class	Dune::Functions::TaylorHoodPreBasis< GV, HI >
	Pre-basis for lowest order Taylor-Hood basis. More...
class	Dune::Functions::Experimental::TransformedIndexPreBasis< RPB, T >
	A pre-basis transforming multi-indices. More...

Typedefs
template<typename GV >
using	Dune::Functions::BSplineBasis = DefaultGlobalBasis< BSplinePreBasis< GV > >
	A global B-spline basis. More...
template<class GV , bool reduced = false, class R = double>
using	Dune::Functions::CubicHermiteBasis = DefaultGlobalBasis< CubicHermitePreBasis< GV, R, reduced > >
	Nodal basis of a scalar cubic Hermite finite element space. More...
template<typename GV , int k, typename R = double>
using	Dune::Functions::HierarchicalLagrangeBasis = DefaultGlobalBasis< HierarchicalLagrangePreBasis< GV, k, R > >
	Basis of a scalar Hierarchical Lagrange finite element space. More...
template<typename GV , int k = -1, typename R = double>
using	Dune::Functions::LagrangeBasis = DefaultGlobalBasis< LagrangePreBasis< GV, k, R > >
	Nodal basis of a scalar k-th-order Lagrangean finite element space. More...
template<typename GV , int k>
using	Dune::Functions::LagrangeDGBasis = DefaultGlobalBasis< LagrangeDGPreBasis< GV, k > >
	Basis of a scalar k-th-order Lagrangean-DG finite element space. More...
template<class GV , class R = double>
using	Dune::Functions::MorleyBasis = DefaultGlobalBasis< MorleyPreBasis< GV, R > >
	Nodal basis of a scalar quadratic Morley finite element space. More...
template<typename GV >
using	Dune::Functions::RannacherTurekBasis = DefaultGlobalBasis< RannacherTurekPreBasis< GV > >
	Rannacher-Turek basis. More...
template<typename GV , int k, typename R = double>
using	Dune::Functions::RefinedLagrangeBasis = DefaultGlobalBasis< RefinedLagrangePreBasis< GV, k, R > >
	Nodal basis of a continuous Lagrange finite-element space on a uniformly refined simplex element. More...
template<typename GV >
using	Dune::Functions::TaylorHoodBasis = DefaultGlobalBasis< TaylorHoodPreBasis< GV > >
	Nodal basis for a lowest order Taylor-Hood Lagrangean finite element space. More...
template<typename GV >
using	Dune::Functions::ReducedCubicHermiteBasis = DefaultGlobalBasis< CubicHermitePreBasis< GV, true > >
	CubicHermite basis. More...

Functions
template<std::size_t k>
auto	Dune::Functions::BasisFactory::brezziDouglasMarini ()
	Create a pre-basis factory that can create a Brezzi-Douglas-Marini pre-basis. More...
auto	Dune::Functions::BasisFactory::bSpline (const std::vector< double > &knotVector, unsigned int order, bool makeOpen=true)
	Create a pre-basis factory that can create a B-spline pre-basis.
template<typename... Args, std::enable_if_t< Concept::isIndexMergingStrategy< typename LastType< Args... >::type >(), int > = 0>
auto	Dune::Functions::BasisFactory::composite (Args &&... args)
	Create a factory builder that can build a CompositePreBasis. More...
template<class ChildPreBasisFactory , class IndexMergingStrategy >
auto	Dune::Functions::BasisFactory::power (ChildPreBasisFactory &&childPreBasisFactory, std::size_t k, const IndexMergingStrategy &)
	Create a pre-basis factory that can build a PowerPreBasis. More...
template<class ChildPreBasisFactory >
auto	Dune::Functions::BasisFactory::power (ChildPreBasisFactory &&childPreBasisFactory, std::size_t k)
	Create a factory builder that can build a PowerPreBasis. More...
template<int k, typename R = double>
auto	Dune::Functions::BasisFactory::hierarchicalLagrange ()
	A factory that can create a HierarchicalLagrange pre-basis. More...
template<std::size_t k, typename R = double>
auto	Dune::Functions::BasisFactory::lagrange ()
	Create a pre-basis factory that can create a Lagrange pre-basis. More...
template<typename R = double>
auto	Dune::Functions::BasisFactory::lagrange (int order)
	Create a pre-basis factory that can create a Lagrange pre-basis with a run-time order. More...
template<std::size_t k>
auto	Dune::Functions::BasisFactory::lagrangeDG ()
	Create a pre-basis factory that can create a LagrangeDG pre-basis. More...
template<std::size_t kind, std::size_t order, typename Range = double>
auto	Dune::Functions::BasisFactory::nedelec ()
	Create a pre-basis factory that can create a Nédélec pre-basis. More...
template<class RawPreBasisIndicator , class PIS >
auto	Dune::Functions::BasisFactory::Experimental::periodic (RawPreBasisIndicator &&rawPreBasisIndicator, PIS &&periodicIndexSet)
	Create a pre-basis factory that can create a periodic pre-basis. More...
template<std::size_t k, class ChildPreBasisFactory , class IndexMergingStrategy >
auto	Dune::Functions::BasisFactory::power (ChildPreBasisFactory &&childPreBasisFactory, const IndexMergingStrategy &)
	Create a pre-basis factory that can build a PowerPreBasis. More...
template<std::size_t k, class ChildPreBasisFactory >
auto	Dune::Functions::BasisFactory::power (ChildPreBasisFactory &&childPreBasisFactory)
	Create a factory builder that can build a PowerPreBasis. More...
template<class Dummy = void>
auto	Dune::Functions::BasisFactory::rannacherTurek ()
	Create a pre-basis factory that can create a Rannacher-Turek pre-basis.
template<std::size_t k>
auto	Dune::Functions::BasisFactory::raviartThomas ()
	Create a pre-basis factory that can create a Raviart-Thomas pre-basis. More...
template<int k, typename R = double>
auto	Dune::Functions::BasisFactory::refinedLagrange ()
	Create a pre-basis factory that can create a RefinedLagrange pre-basis. More...
auto	Dune::Functions::BasisFactory::taylorHood ()
	Create a pre-basis factory that can create a Taylor-Hood pre-basis.
template<class Dummy = void>
auto	Dune::Functions::BasisFactory::cubicHermite ()
	Create a pre-basis factory that can create a CubicHermite pre-basis.
template<class Dummy = void>
auto	Dune::Functions::BasisFactory::reducedCubicHermite ()
	Create a pre-basis factory that can create a CubicHermite pre-basis.

Detailed Description

Implementations of the GlobalBasis concept.

Typedef Documentation

BSplineBasis

template<typename GV >

using Dune::Functions::BSplineBasis = typedef DefaultGlobalBasis<BSplinePreBasis<GV> >

A global B-spline basis.

Template Parameters

GV	The GridView that the space is defined on

CubicHermiteBasis

template<class GV , bool reduced = false, class R = double>

using Dune::Functions::CubicHermiteBasis = typedef DefaultGlobalBasis<CubicHermitePreBasis<GV> >

Nodal basis of a scalar cubic Hermite finite element space.

CubicHermite basis.

Note

This only works for simplex grids. The Hermite basis is only implemented for degree 3.

The Hermite Finite element has the following properties:

• Its global space is in \( C^1 \) if 1d otherwise in \( H^1 \).
• The reduced version (only 2d) is part of the Discrete Kirchhoff Triangle.
• Its interpolation evaluates derivatives, i.e. you cannot interpolate into a lambda function.
• Strongly enforcing boundary conditions is not as simple as with langrange bases
• It global space is not nested, i.e. the space on a refined grid is not a subspace of the space on the coarser grid. All arguments passed to the constructor will be forwarded to the constructor of CubicHermitePreBasis.

Template Parameters

GV	The GridView that the space is defined on
reduced	whether to use the reduced Hermite finite element (only in 2d).
R	The range type of the local basis
GV	The GridView that the space is defined on

HierarchicalLagrangeBasis

template<typename GV , int k, typename R = double>

using Dune::Functions::HierarchicalLagrangeBasis = typedef DefaultGlobalBasis<HierarchicalLagrangePreBasis<GV, k, R> >

Basis of a scalar Hierarchical Lagrange finite element space.

Template Parameters

GV	The GridView that the space is defined on
k	The order of the basis (0 < k <= 2)
R	The range type of the local basis

Note

currently only supports simplex grids

LagrangeBasis

template<typename GV , int k = -1, typename R = double>

using Dune::Functions::LagrangeBasis = typedef DefaultGlobalBasis<LagrangePreBasis<GV, k, R> >

Nodal basis of a scalar k-th-order Lagrangean finite element space.

Note

This only works for certain grids. The following restrictions hold

• If k is no larger than 2, then the grids can have any dimension
• If k is larger than 3 then the grid must be two-dimensional
• If k is 3, then the grid can be 3d if it is a simplex grid

All arguments passed to the constructor will be forwarded to the constructor of LagrangePreBasis.

Warning

The implementation of the basis with run-time order order uses the LagrangeFiniteElement implementation of dune-localfunctions, which is known to violate strict-aliasing rules (see https://gitlab.dune-project.org/core/dune-localfunctions/issues/14) Keep this in mind if ever you experience difficult-to-explain crashes or wrong results.

Template Parameters

GV	The GridView that the space is defined on
k	The order of the basis; -1 means 'order determined at run-time'
R	The range type of the local basis

LagrangeDGBasis

template<typename GV , int k>

using Dune::Functions::LagrangeDGBasis = typedef DefaultGlobalBasis<LagrangeDGPreBasis<GV, k> >

Basis of a scalar k-th-order Lagrangean-DG finite element space.

Template Parameters

GV	The GridView that the space is defined on
k	The order of the basis

MorleyBasis

template<class GV , class R = double>

using Dune::Functions::MorleyBasis = typedef DefaultGlobalBasis<MorleyPreBasis<GV, R> >

Nodal basis of a scalar quadratic Morley finite element space.

Note

This only works for simplex grids. The Morley basis is only implemented for degree 2.

The Morley Finite element has the following properties:

• Its global space is not in \( H^1 \) but can be used for nonconforming method in \( H^2\).
• Its interpolation evaluates derivatives, i.e. you cannot interpolate into a lambda function.
• Strongly enforcing boundary conditions is not as simple as with langrange bases.
• It global space is not nested, i.e. the space on a refined grid is not a subspace of the space on the coarser grid. All arguments passed to the constructor will be forwarded to the constructor of MorleyPreBasis.

Template Parameters

GV	The GridView that the space is defined on
R	The range type of the local basis

RannacherTurekBasis

template<typename GV >

using Dune::Functions::RannacherTurekBasis = typedef DefaultGlobalBasis<RannacherTurekPreBasis<GV> >

Rannacher-Turek basis.

These are Crouzeix-Raviart-elements for quadrilateral elements. See Rolf Rannacher and Stefan Turek. Simple nonconforming quadrilateral Stokes element. Numerical Methods for Partial Differential Equations, 8:97–111, 1992.

Template Parameters

GV	The GridView that the space is defined on

ReducedCubicHermiteBasis

template<typename GV >

using Dune::Functions::ReducedCubicHermiteBasis = typedef DefaultGlobalBasis<CubicHermitePreBasis<GV, true> >

CubicHermite basis.

Template Parameters

GV	The GridView that the space is defined on

RefinedLagrangeBasis

template<typename GV , int k, typename R = double>

using Dune::Functions::RefinedLagrangeBasis = typedef DefaultGlobalBasis<RefinedLagrangePreBasis<GV,k,R> >

Nodal basis of a continuous Lagrange finite-element space on a uniformly refined simplex element.

Note

This only works for simplex grids.

All arguments passed to the constructor will be forwarded to the constructor of RefinedLagrangeBasis.

Template Parameters

GV	The GridView that the space is defined on
k	The order of the basis
R	The range type of the local basis

TaylorHoodBasis

template<typename GV >

using Dune::Functions::TaylorHoodBasis = typedef DefaultGlobalBasis<TaylorHoodPreBasis<GV> >

Nodal basis for a lowest order Taylor-Hood Lagrangean finite element space.

Template Parameters

GV	The GridView that the space is defined on.

Note

This mainly serves as an example, since you can construct a basis with the same functionality manually using

static const int k = 1;
auto taylorHoodBasis = makeBasis(
  gridView,
  composite(
    power<dim>(
      lagrange<k+1>(),
      flatInterleaved()),
    lagrange<k>()
  ));

Function Documentation

brezziDouglasMarini()

template<std::size_t k>

auto Dune::Functions::BasisFactory::brezziDouglasMarini

(

)

Create a pre-basis factory that can create a Brezzi-Douglas-Marini pre-basis.

Template Parameters

k	Order of the Brezzi-Douglas-Marini element

composite()

template<typename... Args, std::enable_if_t< Concept::isIndexMergingStrategy< typename LastType< Args... >::type >(), int > = 0>

auto Dune::Functions::BasisFactory::composite

(

Args &&...

args

)

Create a factory builder that can build a CompositePreBasis.

Template Parameters

Args	Types of child factory builders and IndexMergingStrategy type

Parameters

args	Child factory builder objects and an IndexMergingStrategy

This is the overload used if the last argument is an IndexMergingStrategy.

Template Parameters

Args	Types of child factory builders

Parameters

args	Child factory builder objects

This is the overload used if no IndexMergingStrategy is supplied. In this case the BasisFactory::BlockedLexicographic strategy is used.

References Dune::applyPartial().

hierarchicalLagrange()

template<int k, typename R = double>

auto Dune::Functions::BasisFactory::hierarchicalLagrange

(

)

A factory that can create a HierarchicalLagrange pre-basis.

Template Parameters

k	The polynomial order of the ansatz functions (0 < k <= 2)
R	The range field-type of the local basis

lagrange() [1/2]

template<std::size_t k, typename R = double>

auto Dune::Functions::BasisFactory::lagrange

(

)

Create a pre-basis factory that can create a Lagrange pre-basis.

Template Parameters

k	The polynomial order of the ansatz functions; -1 means 'order determined at run-time'
R	The range type of the local basis

References Dune::Functions::BasisFactory::lagrange().

Referenced by Dune::Functions::BasisFactory::lagrange().

lagrange() [2/2]

template<typename R = double>

auto Dune::Functions::BasisFactory::lagrange

(

int

order

)

Create a pre-basis factory that can create a Lagrange pre-basis with a run-time order.

Template Parameters

R	The range type of the local basis

References Dune::Functions::BasisFactory::lagrange().

lagrangeDG()

template<std::size_t k>

auto Dune::Functions::BasisFactory::lagrangeDG

(

)

Create a pre-basis factory that can create a LagrangeDG pre-basis.

Template Parameters

k	The polynomial order of the ansatz functions

nedelec()

template<std::size_t kind, std::size_t order, typename Range = double>

auto Dune::Functions::BasisFactory::nedelec

(

)

Create a pre-basis factory that can create a Nédélec pre-basis.

Template Parameters

kind	Kind of the Nédélec element (1 or 2)
order	Order of the Nédélec element (lowest order is '1')
Range	Number type used for shape function values

periodic()

template<class RawPreBasisIndicator , class PIS >

auto Dune::Functions::BasisFactory::Experimental::periodic	(	RawPreBasisIndicator &&	rawPreBasisIndicator,
		PIS &&	periodicIndexSet
	)

Create a pre-basis factory that can create a periodic pre-basis.

Parameters

rawPreBasisIndicator	Object encoding the raw non-periodic basis
periodicIndexSet	A PeriodicIndexSet containing the indices to be identified

The rawPreBasisIndicator can either be a PreBasisFactory, a PreBasis, or a GlobalBasis. In the latter two cases the multi index type of those bases and the periodic basis to be constructed, has to coincide. Both arguments will be copied. Currently only wrapped bases with flat indices are supported.

Examples

recipe-operator-splitting.cc.

References Dune::Functions::BasisFactory::Experimental::periodic().

Referenced by Dune::Functions::BasisFactory::Experimental::periodic().

power() [1/4]

template<std::size_t k, class ChildPreBasisFactory >

auto Dune::Functions::BasisFactory::power

(

ChildPreBasisFactory &&

childPreBasisFactory

)

Create a factory builder that can build a PowerPreBasis.

Template Parameters

ChildPreBasisFactory

Types of child pre-basis factory

Parameters

childPreBasisFactory

Child pre-basis factory

This overload will select the BasisFactory::BlockedInterleaved strategy.

power() [2/4]

template<std::size_t k, class ChildPreBasisFactory , class IndexMergingStrategy >

auto Dune::Functions::BasisFactory::power	(	ChildPreBasisFactory &&	childPreBasisFactory,
		const IndexMergingStrategy &
	)

Create a pre-basis factory that can build a PowerPreBasis.

Template Parameters

ChildPreBasisFactory	Types of child pre-basis factory
IndexMergingStrategy	An IndexMergingStrategy type

Parameters

childPreBasisFactory	Child pre-basis factory
ims	IndexMergingStrategy to be used

This overload can be used to explicitly supply an IndexMergingStrategy.

power() [3/4]

template<class ChildPreBasisFactory >

auto Dune::Functions::BasisFactory::power	(	ChildPreBasisFactory &&	childPreBasisFactory,
		std::size_t	k
	)

Create a factory builder that can build a PowerPreBasis.

Template Parameters

ChildPreBasisFactory

Types of child pre-basis factory

Parameters

childPreBasisFactory

Child pre-basis factory

This overload will select the BasisFactory::BlockedInterleaved strategy.

power() [4/4]

template<class ChildPreBasisFactory , class IndexMergingStrategy >

auto Dune::Functions::BasisFactory::power	(	ChildPreBasisFactory &&	childPreBasisFactory,
		std::size_t	k,
		const IndexMergingStrategy &
	)

Create a pre-basis factory that can build a PowerPreBasis.

Template Parameters

ChildPreBasisFactory	Types of child pre-basis factory
IndexMergingStrategy	An IndexMergingStrategy type

Parameters

childPreBasisFactory	Child pre-basis factory
ims	IndexMergingStrategy to be used

This overload can be used to explicitly supply an IndexMergingStrategy.

raviartThomas()

template<std::size_t k>

auto Dune::Functions::BasisFactory::raviartThomas

(

)

Create a pre-basis factory that can create a Raviart-Thomas pre-basis.

Template Parameters

k	Order of the Raviart-Thomas element

refinedLagrange()

template<int k, typename R = double>

auto Dune::Functions::BasisFactory::refinedLagrange

(

)

Create a pre-basis factory that can create a RefinedLagrange pre-basis.

Template Parameters

R	The range type of the local basis