Dune::ACFem::P_LaplacianOperatorParts< FunctionSpace > Class Template Reference

The p-Laplacian-model. More...

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

Public Types
enum	StructureFlags
enum	ConstituentFlags
typedef Expression	ExpressionType
	The type of the underlying expression.

Public Member Functions
template<class Entity , class Point >
void	flux (const Entity &entity, const Point &x, const RangeType &value, const JacobianRangeType &jacobian, JacobianRangeType &flux) const
	Evaluate \(A(x, u)\nabla u(x)\) in local coordinates. More...
template<class Entity , class Point >
void	linearizedFlux (const RangeType &uBar, const JacobianRangeType &DuBar, const Entity &entity, const Point &x, const RangeType &value, const JacobianRangeType &jacobian, JacobianRangeType &flux) const
	Evaluate the linearized flux in local coordinates. More...
template<class Entity , class Point >
void	fluxDivergence (const Entity &entity, const Point &x, const RangeType &value, const JacobianRangeType &jacobian, const HessianRangeType &hessian, RangeType &result) 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...
void	setEntity (const Entity &entity) const
bool	setIntersection (const Intersection &intersection) const
void	source (const Entity &entity, const Point &x, const RangeType &value, const JacobianRangeType &jacobian, RangeType &result) const
void	linearizedSource (const RangeType &uBar, const JacobianRangeType &DuBar, const Entity &entity, const Point &x, const RangeType &value, const JacobianRangeType &jacobian, RangeType &result) const
void	robinFlux (const Intersection &intersection, const Point &x, const DomainType &unitOuterNormal, const RangeType &value, RangeType &result) const
void	linearizedRobinFlux (const RangeType &uBar, const Intersection &intersection, const Point &x, const DomainType &unitOuterNormal, const RangeType &value, RangeType &result) const
const ExpressionType &	expression () const
	Return a const reference to the underlying expression.
ExpressionType &	expression ()
	Return a mutable reference to the underlying expression.
ExpressionType	operator* () const
	Return a copy from of the underlying expression.

Detailed Description

template<class FunctionSpace>
class Dune::ACFem::P_LaplacianOperatorParts< FunctionSpace >

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 Enumeration Documentation

ConstituentFlags

enum Dune::ACFem::DefaultOperatorParts::ConstituentFlags

inherited

StructureFlags

enum Dune::ACFem::DefaultOperatorParts::StructureFlags

inherited

Member Function Documentation

linearizedRobinFlux()

void Dune::ACFem::DefaultOperatorParts< Expression >::linearizedRobinFlux ( const RangeType &  uBar, const Intersection &  intersection, const Point &  x, const DomainType &  unitOuterNormal, const RangeType &  value, RangeType &  result  ) const

inlineinherited

linearizedSource()

void Dune::ACFem::DefaultOperatorParts< Expression >::linearizedSource ( const RangeType &  uBar, const JacobianRangeType &  DuBar, const Entity &  entity, const Point &  x, const RangeType &  value, const JacobianRangeType &  jacobian, RangeType &  result  ) const

inlineinherited

robinFlux()

void Dune::ACFem::DefaultOperatorParts< Expression >::robinFlux ( const Intersection &  intersection, const Point &  x, const DomainType &  unitOuterNormal, const RangeType &  value, RangeType &  result  ) const

inlineinherited

setEntity()

void Dune::ACFem::DefaultOperatorParts< Expression >::setEntity ( const Entity &  entity ) const

inlineinherited

setIntersection()

bool Dune::ACFem::DefaultOperatorParts< Expression >::setIntersection ( const Intersection &  intersection ) const

inlineinherited

source()

void Dune::ACFem::DefaultOperatorParts< Expression >::source ( const Entity &  entity, const Point &  x, const RangeType &  value, const JacobianRangeType &  jacobian, RangeType &  result  ) const

inlineinherited

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

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