Dune::ACFem::PDEModel::P_LaplacianModel< FunctionSpace, PField > Class Template Reference

The p-Laplacian-model. More...

#include <dune/acfem/models/modules/plaplacianmodel.hh>

Collaboration diagram for Dune::ACFem::PDEModel::P_LaplacianModel< FunctionSpace, PField >:

Public Member Functions
auto	flux (const JacobianRangeType &jacobian) const
	Evaluate \(A(x, u)\nabla u(x)\) in local coordinates. More...
auto	linearizedFlux (const JacobianRangeType &DuBar, const JacobianRangeType &jacobian) const
	Evaluate the linearized flux in local coordinates. More...
auto	fluxDivergence (const JacobianRangeType &jacobian, const HessianRangeType &hessian) const
	Compute the point-wise value of the flux-part of the operator, meaning the part of the differential operator which is multiplied by the derivative of the test function. More...

FunctionSpaceTypes
Forward some basic type from the supplied function-spaces to the model class. Note that FunctionSpace is defined as a shortcut for RangeFunctionSpace in order to simplify the common case DomainFunctionSpace == RangeFunctionSpace.
using	HessianRangeSelector = typename std::conditional< std::is_convertible< typename FunctionSpace::RangeType, typename FunctionSpace::HessianRangeType >::value, Fem::ExplicitFieldVector< typename FunctionSpace::HessianRangeType::value_type, FunctionSpace::dimRange >, typename FunctionSpace::HessianRangeType >::type
	The type returned by classifyBoundary().
using	RangeFunctionSpaceType = FunctionSpace
	The type returned by classifyBoundary().
using	DomainFunctionSpaceType = FunctionSpace
	The type returned by classifyBoundary().
using	FunctionSpaceType = RangeFunctionSpaceType
	The type returned by classifyBoundary().
using	DomainType = typename FunctionSpaceType::DomainType
	The type returned by classifyBoundary().
using	RangeType = typename FunctionSpaceType::RangeType
	The type returned by classifyBoundary().
using	JacobianRangeType = typename FunctionSpaceType::JacobianRangeType
	The type returned by classifyBoundary().
using	HessianRangeType = HessianRangeSelector< FunctionSpaceType >
	The type returned by classifyBoundary().
using	DomainFieldType = typename FunctionSpaceType::DomainFieldType
	The type returned by classifyBoundary().
using	RangeFieldType = typename FunctionSpaceType::RangeFieldType
	The type returned by classifyBoundary().
using	DomainRangeType = typename DomainFunctionSpaceType::RangeType
	The type returned by classifyBoundary().
using	DomainDomainType = typename DomainFunctionSpaceType::DomainType
	The type returned by classifyBoundary().
using	DomainJacobianRangeType = typename DomainFunctionSpaceType::JacobianRangeType
	The type returned by classifyBoundary().
using	DomainHessianRangeType = HessianRangeSelector< DomainFunctionSpaceType >
	The type returned by classifyBoundary().
using	DomainDomainFieldType = typename DomainFunctionSpaceType::DomainFieldType
	The type returned by classifyBoundary().
using	DomainRangeFieldType = typename DomainFunctionSpaceType::RangeFieldType
	The type returned by classifyBoundary().
using	RangeRangeType = typename RangeFunctionSpaceType::RangeType
	The type returned by classifyBoundary().
using	RangeDomainType = typename RangeFunctionSpaceType::DomainType
	The type returned by classifyBoundary().
using	RangeJacobianRangeType = typename RangeFunctionSpaceType::JacobianRangeType
	The type returned by classifyBoundary().
using	RangeHessianRangeType = HessianRangeSelector< RangeFunctionSpaceType >
	The type returned by classifyBoundary().
using	RangeDomainFieldType = typename RangeFunctionSpaceType::DomainFieldType
	The type returned by classifyBoundary().
using	RangeRangeFieldType = typename RangeFunctionSpaceType::RangeFieldType
	The type returned by classifyBoundary().
using	BoundaryConditionsType = std::pair< bool, std::bitset< dimRange > >
	The type returned by classifyBoundary().
void	bind (const Entity &entity)
	Bind to the given entity. More...
void	unbind ()
	Unbind from the previously bound entity. More...
auto	classifyBoundary (const Intersection &intersection)
	Bind to the given intersection and classify the components w.r.t. More...
static constexpr int	dimDomain
	The type returned by classifyBoundary().
static constexpr int	dimRange
	The type returned by classifyBoundary().
static constexpr int	domainDimDomain
	The type returned by classifyBoundary().
static constexpr int	domainDimRange
static constexpr int	rangeDimDomain
	The type returned by classifyBoundary().
static constexpr int	rangeDimRange

Detailed Description

template<class FunctionSpace, class PField>
class Dune::ACFem::PDEModel::P_LaplacianModel< FunctionSpace, PField >

The p-Laplacian-model.

The flux-terms has the form

\[ \int_\Omega |\nabla U|^{p-2}\,\nabla U : \nabla \phi\quad\forall \phi \]

where \(U\) is the unknown und \(ß\phi\) denotes the test functions. This is the formal first variation of the functional

\[ U\mapsto \int_\Omega |\nabla U|^p = \int_\Omega (\nabla U : \nabla U)^{p/2}. \]

FIXME: use rather the following for the vector valud case?

\[ U\mapsto \int_\Omega \sum_{i=1}^r (\nabla U_i\cdot\nabla U_i)^{p/2}. \]

The second form yields a component-wise uncoupled p-Laplacian, the first yields a coupled version. We implement here the first, but this may not be what you want.

Bug:

This is totally untested. Cross check before using and then remove the comment.

Member Function Documentation

bind()

void Dune::ACFem::ModelBase< FunctionSpace , FunctionSpace >::bind ( const Entity &  entity )

inlineinherited

Bind to the given entity.

Parameters

[in]

entity

The entity to bind to.

Warning

Calling any other method without first binding the model results in undefined behaviour.

Models needing this method need to reimplement it. This is just here to obey the "bindable" interface of Dune::Fem.

classifyBoundary()

auto Dune::ACFem::ModelBase< FunctionSpace , FunctionSpace >::classifyBoundary ( const Intersection &  intersection )

inlineinherited

Bind to the given intersection and classify the components w.r.t.

to the kind of applicable boundary conditions.

Warning

Note that prior to calling this function the model has to be bound to the inside entity of the given intersection. Failing to do so generates undefined behaviour.

The result of calling the other boundary related methods without binding to an intersection is undefined.

If RESULT.first is false, then the result of calling any of the other boundary related functions is undefined. Philosophically, they should return 0 in this case, but in order to have decent performance they give a damn and just don't care.

If RESULT.first is true, then still you cannot rely on user-friendly behaviour:

• only if the respective bit of RESULT.second is set to 1, then the Dirichlet value in this compoment is well-defined.
• only if the respective bit of RESULT.second is set to 0, then the Robin value in this component is well defined.

Parameters

[in]

intersection

The intersection to bind to.

Returns

A tuple. First component is a bool which is true iff any of the boundary related data functions would result in non trivial results. Second component is a bitset of size dimRange which is true if the given component of the system is subject to Dirichlet boundary conditions and false if it is subject to Robin or Neumann boundary conditions. If first is false then the contents of the bitset is undefined.

unbind()

void Dune::ACFem::ModelBase< FunctionSpace , FunctionSpace >::unbind ( )

inlineinherited

Unbind from the previously bound entity.

Warning

Calling this method on an unbound model may cause undefined behaviour.

Models needing this method need to reimplement it. This is just here to obey the "bindable" interface of Dune::Fem.

Member Data Documentation

domainDimRange

constexpr int Dune::ACFem::ModelBase< FunctionSpace , FunctionSpace >::domainDimRange

staticconstexprinherited

rangeDimRange

constexpr int Dune::ACFem::ModelBase< FunctionSpace , FunctionSpace >::rangeDimRange

staticconstexprinherited

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

• dune/acfem/models/modules/plaplacianmodel.hh