Dune::ACFem::EllipticOperator< OperatorParts, DomainFunction, RangeFunction, Constraints, QuadratureTraits > Class Template Reference

A class defining an elliptic operator. More...

#include <dune/acfem/operators/ellipticoperator.hh>

Public Member Functions
	EllipticOperator (const OperatorPartsType &parts, const ConstraintsOperatorType &constraints=ConstraintsOperatorType())
	contructor
virtual void	operator() (const DomainFunctionType &u, RangeFunctionType &w) const
	application operator, works "on the fly"
virtual bool	symmetric () const
	linear operator interface

Detailed Description

template<class OperatorParts, class DomainFunction, class RangeFunction = DomainFunction, class Constraints = EmptyBlockConstraints<RangeFunction>, template< class > class QuadratureTraits = DefaultQuadratureTraits>
class Dune::ACFem::EllipticOperator< OperatorParts, DomainFunction, RangeFunction, Constraints, QuadratureTraits >

A class defining an elliptic operator.

A class defining an elliptic differential operator \(\Phi\) of second order of the following type:

\[ \Phi: g+X \rightarrow X^\ast: u\mapsto \Phi(u). \]

\(\Phi\) – in general – maps an affine linear space (e.g. shifted by Dirichlet boundary values) to the dual of the attached vector space. The idea is that one always – also for linear problems – searches for zeros of \(\Phi\), i.e. the goal is

Find \(u\in g+X\) with \(\Phi(u) = 0\in X^\ast\).

The strong and weak form of the problem class this construct is intended for reads then

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]	DiscreteFunction	The underlying discrete function space.
[in]	Model	A class defining the differential operator. See class ModelInterface for the interface definition. The module Model Building Blocks provides some standard models which can be chained together by means of expression-templates in natural notation. EllipticModel contains an implementation based on dynamic polymorphism; it is meant for demonstration purposes only.
[in]	Constraints	A class modelling the affine shift, for example by interpolating Dirichlet values into the DOFs belonging to the boundary. How this is done is left to the constraints-implementation. See BlockConstraintsOperatorInterface and DirichletConstraints.

Todo:

Does not allow for mapping between different function spaces. This could become important for systems like Stokes.

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

• dune/acfem/operators/ellipticoperator.hh