Interface class for second order elliptic models. More...

#include <dune/acfem/models/operatorparts/operatorparts.hh>

Public Types
enum	StructureFlags { isLinear = TraitsType::isLinear , isSymmetric = TraitsType::isSymmetric , isSemiDefinite = TraitsType::isSemiDefinite }
	Static flags for the overall structure of the operator. More...

enum	ConstituentFlags
	Provide information about the constituents of the model. More...

Public Member Functions
template<class Entity >
void	setEntity (const Entity &entity) const
	Per entity initialization, if that is needed. More...

template<class Intersection >
bool	setIntersection (const Intersection &intersection) const
	Per-intersection initialization for the boundary contributions. More...

std::string	name () const
	Print a descriptive name for debugging and output.

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	source (const Entity &entity, const Point &x, const RangeType &value, const JacobianRangeType &jacobian, RangeType &result) const
	The zero-order term as function of local coordinates. More...

template<class Entity , class Point >
void	linearizedSource (const RangeType &uBar, const JacobianRangeType &DuBar, const Entity &entity, const Point &x, const RangeType &value, const JacobianRangeType &jacobian, RangeType &result) const
	The linearized source term as function of local coordinates. More...

template<class Intersection , class Point >
void	robinFlux (const Intersection &intersection, const Point &x, const DomainType &unitOuterNormal, const RangeType &value, RangeType &result) const
	The non-linearized Robin-type flux term. More...

template<class Intersection , class Point >
void	linearizedRobinFlux (const RangeType &uBar, const Intersection &intersection, const Point &x, const DomainType &unitOuterNormal, const RangeType &value, RangeType &result) const
	The linearized Robin-type flux term. 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...

Detailed Description

template<class PartsImpl>
class Dune::ACFem::OperatorPartsInterface< PartsImpl >

Interface class for second order elliptic models.

The ModelInterface defines all the different parts needed to compute the weak form of a second order elliptic operator of the form

\[ \begin{split} -\nabla\cdot (A(x, u, \nabla u)\,\nabla u) + \nabla\cdot(b(x, u)\,u) + c(x, u)\,u &= f(x) + \Psi\quad \text{ in }\Omega,\\ u &= g_D \text{ on }\Gamma_D,\\ (A(x, u, \nabla u)\nabla u)\cdot\nu + \alpha(x, u)\,u &= g_N \text{ on }\Gamma_R,\\ (A(x, u, \nabla u)\nabla u)\cdot\nu &= g_N \text{ on }\Gamma_N,\\ \end{split} \]

In particular, $\Psi\in H^{-1}$ models a functional, $f\in L^2$ is the usual "right hand side".

The methods in this class provide all factors of the integrands not involving test functions. The multiplication by the test-functions is added in the class EllipticOperator. The weak formulation then reads:

\[ \int_\Omega (A(x,u,\nabla u)\,\nabla u)\cdot\nabla\phi\,dx + \int_\Omega (\nabla\cdot(b(x, u)\,u) + c(x, u)\,u)\,\phi\,dx + \int_{\Gamma_R} \alpha(x, u)\,u\,\phi\,do - \int_{\Gamma_N} g_N\,\phi\,do - \int_\Omega f(x)\,\phi\,dx - \langle\Psi,\,\phi\rangle = 0. \]

As the ModelInterface is potentially non-linear one has the option to leave the ModelInterface::BulkForcesFunctionType and ModelInterface::NeumannBoundaryFunctionType at zero and instead stuff the "right-hand-side" functions into the ModelInterface::robinFlux() and ModelInterface::source() methods. In principle even the functional $\Psi$ could be expressed by a $L^2$ scalar-product (don't shout at me: as the FEM-space is finite dimensional this is of course possible and one way to implement such a functional).

Nevertheless the different forces and other right-hand-side components are there and will be used by EllipticFemScheme and ParabolicFemScheme. Generally, the implementation checks for zero-objects (see ZeroExpression, ZeroGridFunction, ZeroFunctional, ZeroModel); the decision about which component has to be taken into account is taken at compile-time.

Parameters

[in] PartsImpl The implementation.

Bug:: The setEntity() method is evil, because it in principle prevents models from being tagged as "const", or is the reason for the evil use of the "mutable" keyword in several places. Way out would be an "OperatorGerm"-class which is constructed from a given entity and/or intersection.

Todo:: In order to support systems there should be a DomainFunctionSpace (ansatz-functions) and a RangeFunctionSpace (test-functions). Example would be velocity-pressure for mixed discretizations of Stokes-like problems. Although it would be possible to stuff such a problem into the existing model-problem this does not feel like being desirable.

Member Enumeration Documentation

◆ ConstituentFlags

template<class PartsImpl >

enum Dune::ACFem::OperatorPartsInterface::ConstituentFlags

Provide information about the constituents of the model.

All other components (like forces, boundary values) can be identified by looking at the data type.

◆ StructureFlags

template<class PartsImpl >

enum Dune::ACFem::OperatorPartsInterface::StructureFlags

Static flags for the overall structure of the operator.

Enumerator
isLinear	Define to true for the affine-linear case.
isSymmetric	Define to true for the symmetric case.
isSemiDefinite	Define to true for the non-indefinite case (non trivial kernel is allowed).

Member Function Documentation

◆ flux()

template<class PartsImpl >

template<class Entity , class Point >

void Dune::ACFem::OperatorPartsInterface< PartsImpl >::flux	(	const Entity &	entity,
		const Point &	x,
		const RangeType &	value,
		const JacobianRangeType &	jacobian,
		JacobianRangeType &	flux
	)		const

inline

Evaluate $A(x, u)\nabla u(x)$ in local coordinates.

This can be interpreted as a diffusive "flux", at least when restricted to some surface. $\bar u$ denotes the point of linearization for non-linear problems.

Parameters

[in]	entity	The mesh element.
[in]	x	The point of evaluation, local coordinates.
[in]	value	To allow integration by parts for first order terms and in preparation for non-linear models.
[in]	jacobian	The value of $\nabla\bar\psi$.
[out]	flux	The result of the computation.

Note: "flux" in this sense simply means something which is multiplied by the jacobians of the test-function and covers "diffusive fluxes" as well as advective fluxes.

References Dune::ACFem::asImp().

Referenced by Dune::ACFem::OperatorPartsInterface< PartsImpl >::linearizedFlux().

◆ fluxDivergence()

template<class PartsImpl >

template<class Entity , class Point >

void Dune::ACFem::OperatorPartsInterface< PartsImpl >::fluxDivergence	(	const Entity &	entity,
		const Point &	x,
		const RangeType &	value,
		const JacobianRangeType &	jacobian,
		const HessianRangeType &	hessian,
		RangeType &	result
	)		const

inline

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.

Primarily useful for residual error estimators.

Parameters

[in]	entity	The mesh element.
[in]	x	The point of evaluation, local coordinates.
[in]	value	In preparation for non-linear problems.
[in]	jacobian	The value of $\nabla u$.
[in]	hessian	The value of $D^2u$.
[out]	result	The result of the computation.

References Dune::ACFem::asImp().

◆ linearizedFlux()

template<class PartsImpl >

template<class Entity , class Point >

void Dune::ACFem::OperatorPartsInterface< PartsImpl >::linearizedFlux	(	const RangeType &	uBar,
		const JacobianRangeType &	DuBar,
		const Entity &	entity,
		const Point &	x,
		const RangeType &	value,
		const JacobianRangeType &	jacobian,
		JacobianRangeType &	flux
	)		const

inline

Evaluate the linearized flux in local coordinates.

This can be interpreted as a diffusive "flux", at least when restricted to some surface. $\bar u$ denotes the point of linearization for non-linear problems.

Parameters

[in]	uBar	Point of linearization.
[in]	DuBar	Point of linearization.
[in]	entity	The mesh element.
[in]	x	The point of evaluation, local coordinates.
[in]	value	To allow integration by parts for first order terms and in preparation for non-linear models.
[in]	jacobian	The value of $\nabla\bar\psi$.
[out]	flux	The result of the computation.

Note: "flux" in this sense simply means something which is multiplied by the jacobians of the test-function and covers "diffusive fluxes" as well as advective fluxes.

References Dune::ACFem::asImp(), and Dune::ACFem::OperatorPartsInterface< PartsImpl >::flux().

Referenced by Dune::ACFem::DefaultOperatorParts< PartsImpl >::flux().

◆ linearizedRobinFlux()

template<class PartsImpl >

template<class Intersection , class Point >

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

inline

The linearized Robin-type flux term.

Parameters

[in]	uBar	The point of linearization.
[in]	intersection	The current intersection.
[in]	x	The point of evaluation, local coordinates.
[in]	unitOuterNormal	The outer normal (outer with respect to the current entity).
[in]	value	The value of u.
[out]	result	The result of the computation.

References Dune::ACFem::asImp().

Referenced by Dune::ACFem::DefaultOperatorParts< PartsImpl >::robinFlux().

◆ linearizedSource()

template<class PartsImpl >

template<class Entity , class Point >

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

inline

The linearized source term as function of local coordinates.

Note the "source" in this context includes all terms which are multiplied by the value (not the jacobian) of the test-functions und thus may also include first order terms.

Parameters

[in]	uBar	The point of linearization.
[in]	DuBar	The jacobian at the point of linearization.
[in]	entity	The mesh element.
[in]	x	The point of evaluation, local coordinates.
[in]	value	The value of u.
[in]	jacobian	The jacobian of u.
[out]	result	The result of the computation.

References Dune::ACFem::asImp().

Referenced by Dune::ACFem::DefaultOperatorParts< PartsImpl >::source().

◆ robinFlux()

template<class PartsImpl >

template<class Intersection , class Point >

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

inline

The non-linearized Robin-type flux term.

This is "@f$\alpha@f$" from the Robin boundary condition.

Parameters

[in]	intersection	The current intersection.
[in]	x	The point of evaluation, local coordinates.
[in]	unitOuterNormal	The outer normal (outer with respect to the current entity).
[in]	value	The value of u.
[out]	result	The result of the computation.

References Dune::ACFem::asImp().

◆ setEntity()

template<class PartsImpl >

template<class Entity >

void Dune::ACFem::OperatorPartsInterface< PartsImpl >::setEntity ( const Entity & entity ) const

inline

Per entity initialization, if that is needed.

Useful for constructing LocalFunction instances if the coefficients depend on discrete data, for example. A cleaner solution would be to provide a "LocalOperatorGerm" class which would be constructed from en EntityType in the same way as the Fem::LocalFunction is constructed from an entity (and a grid-function). OHTO, this would introduce yet another data-structure ...

References Dune::ACFem::asImp().

◆ setIntersection()

template<class PartsImpl >

template<class Intersection >

bool Dune::ACFem::OperatorPartsInterface< PartsImpl >::setIntersection ( const Intersection & intersection ) const

inline

Per-intersection initialization for the boundary contributions.

Several parts of the boundary may carry different Robin-kind boundary conditions. If necessary, per-intersection initializations can be performed here.

Parameters

[in] intersection The Intersection implementation for the underlying grid.

Returns: The function must return true if Robin-boundary conditions apply for intersection.

Note: Potential Dirichlet boundary conditions can be ignored here, the higher level code will only try to assemble boundary contributions if Dirichlet-conditions do not apply. It is therefore safe to constantly return true if there is only a split between a Dirichlet boundary part and a Robin boundary part. However, for more complicated cases it is vital that robinFlux() and linearizedRobinFlux() evaluate to zero if setIntersection() return false in order for ModelExpressions to work correctly,

References Dune::ACFem::asImp().

◆ source()

template<class PartsImpl >

template<class Entity , class Point >

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

inline

The zero-order term as function of local coordinates.

Has to evaluate $\nabla\cdot(b(x,u)\,u) + c(x, u)\,u$.

Parameters

[in]	entity	The mesh element.
[in]	x	The point of evaluation, local coordinates.
[in]	value	The value of u.
[in]	jacobian	The jacobian of u.
[out]	result	The result of the computation.

References Dune::ACFem::asImp().

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

dune/acfem/models/operatorparts/operatorparts.hh

DUNE Distributed and Unified Numerics Environment

DUNE-ACFEM (2.5.1)

Public Types

Public Member Functions

Detailed Description

template<class PartsImpl>
class Dune::ACFem::OperatorPartsInterface< PartsImpl >

Member Enumeration Documentation

◆ ConstituentFlags

◆ StructureFlags

Member Function Documentation

◆ flux()

◆ fluxDivergence()

◆ linearizedFlux()

◆ linearizedRobinFlux()

◆ linearizedSource()

◆ robinFlux()

◆ setEntity()

◆ setIntersection()

◆ source()

Public Types

Public Member Functions

Detailed Description

template<class PartsImpl> class Dune::ACFem::OperatorPartsInterface< PartsImpl >

Member Enumeration Documentation

◆ ConstituentFlags

◆ StructureFlags

Member Function Documentation

◆ flux()

◆ fluxDivergence()

◆ linearizedFlux()

◆ linearizedRobinFlux()

◆ linearizedSource()

◆ robinFlux()

◆ setEntity()

◆ setIntersection()

◆ source()

template<class PartsImpl>
class Dune::ACFem::OperatorPartsInterface< PartsImpl >