Given a elliptic equation: More...

#include <implicitResidual.hh>

Inheritance diagram for estimator::ImplicitResidual2D< F >:

Public Member Functions
void	addBoundaryData (const concepts::Set< uint > &nSet, const concepts::ElementFormula< F > &g)
	Add boundary informations (i.e. More...

void	addBoundaryData (const concepts::Set< uint > &rSet, const concepts::ElementFormula< F > &h1, const concepts::ElementFormula< F > &h2)
	Add boundary informations (i.e. More...

void	addLhs (concepts::BilinearForm< F > &bform, Real w=1.0)

void	addRhs (concepts::LinearForm< F > &lform, Real w=1.0)

virtual ImplicitResidual2D< F > *	clone () const

void	compute ()

	ImplicitResidual2D (const concepts::SpaceOnCells< F > &spc, const concepts::Vector< F > &sol, const concepts::ElementFormula< F > *a=0)
	Constructor of the 2d implicit residual a posteriori Error estimator. More...

virtual const Real	operator() () const

void	setErrorNorm (concepts::ElementFunction< F > &func, const concepts::ElementFormula< F > *w=0, Real s=1.0)

Protected Member Functions
virtual std::ostream &	info (std::ostream &os) const

Protected Attributes
concepts::Set< uint >	eAttrbs_

concepts::Sequence< const concepts::ElementFormula< F > * >	g_

concepts::Sequence< concepts::Set< uint > >	gAttrbs_

Private Member Functions
virtual void	computeError_ ()

Private Attributes
const concepts::ElementFormula< F > *	a_

concepts::Sequence< concepts::BilinearForm< F > * >	bforms_

concepts::Sequence< concepts::LinearForm< F > * >	lforms_

concepts::Sequence< Real >	lws_

concepts::Sequence< uint >	norm_d_

concepts::Sequence< concepts::ElementFunction< F > * >	norm_fs_

concepts::Sequence< Real >	norm_s_

concepts::Sequence< const concepts::ElementFormula< F > * >	norm_ws_

const concepts::Vector< F > &	sol_

concepts::Sequence< Real >	ws_

Detailed Description

template<class F>
class estimator::ImplicitResidual2D< F >

Given a elliptic equation:

     Lu = f  in \Omega
    u = 0  on \Gamma_D  (Dirichletboundary)

\partial_n u = g on \Gamma_N (Neumannboundary)

Let highest order operator be given as div(alpha(x,y)*grad(.)), so consider isotropic case.

For more general case div ( IK * grad(.) ), one may think of a diffusion-tensor IK, this class is not builded up so far. MatrixFormulas will be needed. We just consider the special case IK = I*alpha so far.

Given a FEM solution u_h, the error estimator local on a element K becomes

eta_K^2 := w_K^2*||f-Lu_h||_L^2(K)^2

sum\limits_{E \subset \partial K\ w_E*||[[n^T * alpha grad u_h]]_K||_L^2(E)^2

where [[.]]_K is defined as [[.]] local on a element K where

 { [n^T * alpha grad u_h]     on inner edges E (i.e two underylying Elements)

[[v]] = { g - [n^T * alpha grad u_h] on neumann edges E (one underlying Element) { 0 on dirichlet boundary

while [.] denotes the jump and grad_n the normal derivative.

As weight w_K we may choose

w_K= diam(K)
w_K= diam(K)/(1+p_K) where p_K is minimum polynomial degree along one direction.

For weight w_E we may choose

w_E = diam(E)/2
w_E= diam(E)/2*(1+p_E) where p_E is polynomial degree along the Edge E.

The second weights are motivated from : An hp-Efficient Residual-Based A Posteriori Error Estimator for Maxwell’s Equations, Markus Bürg

and the inverse polynomial Interpolation estimates within. They may have to be taken with care for reliably error estimator, this is not checked.

We assume that all inner edges have attribute zero and no boundary edge has attribute zero.

Author: Robert Gruhlke, 2014

Definition at line 215 of file implicitResidual.hh.

Constructor & Destructor Documentation

◆ ImplicitResidual2D()

template<class F >

estimator::ImplicitResidual2D< F >::ImplicitResidual2D	(	const concepts::SpaceOnCells< F > &	spc,
		const concepts::Vector< F > &	sol,
		const concepts::ElementFormula< F > *	a = `0`
	)

inline

Constructor of the 2d implicit residual a posteriori Error estimator.

Parameters

spc	The FEM space
sol	the solution vector representing the FEM solution
wRes	The weak residual as linearform, e.g. int fv - int grad(uh) grad v for laplace
a	The coefficient function in the highest derivative operator div ( a * grad(.)) if no function `a` is given, by default `a` is const and therefore set to ONE. this is needed to define the local neumannproblems

Definition at line 234 of file implicitResidual.hh.

Member Function Documentation

◆ addBoundaryData() [1/2]

template<class F >

void estimator::ImplicitResidual< F >::addBoundaryData	(	const concepts::Set< uint > &	nSet,
		const concepts::ElementFormula< F > &	g
	)

inlineinherited

Add boundary informations (i.e.

on point/edge/face) to build boundary residuals on a given requested set of edges through edge attributes.

This applicates i.e for Neumann b.c. :

Let the boundary condition be of the form :

 n^T * [A*Grad(u)] = g

Then the L2 edge residual representation has the form :

$|| g - n_e^T * [A*Grad(u_{hp})] ||_{L^2(e)}$

where n_e is the outward pointing normal vector, that comes from the underlying element that has edge e.

Definition at line 64 of file implicitResidual.hh.

◆ addBoundaryData() [2/2]

template<class F >

void estimator::ImplicitResidual< F >::addBoundaryData	(	const concepts::Set< uint > &	rSet,
		const concepts::ElementFormula< F > &	h1,
		const concepts::ElementFormula< F > &	h2
	)

inlineinherited

Add boundary informations (i.e.

on point/edge/face) to build boundary residuals of the form

$h2 + h1 * u_{hp}$

on a given set of edge attributes, where u_{hp} is the trace of the hp-fem solution on the requested edges. This applicates i.e. for Robin b.c. :

Let the boundary condition be of the form :

n^T * [A*Grad(u)] = h1 * u + h2

Then the L2 edge residual representation has the form :

$|| h1 * u_{hp} + h2 - n_e^T * [A*Grad(u_{hp})] ||_{L^2(e)}$

where n_e is the outward pointing normal vector, that comes from the underlying element that has edge e.

Definition at line 98 of file implicitResidual.hh.

◆ addLhs()

template<class F >

void estimator::ImplicitResidual2D< F >::addLhs	(	concepts::BilinearForm< F > &	bform,
		Real	w = `1.0`
	)

inline

Definition at line 255 of file implicitResidual.hh.

◆ addRhs()

template<class F >

void estimator::ImplicitResidual2D< F >::addRhs	(	concepts::LinearForm< F > &	lform,
		Real	w = `1.0`
	)

inline

Definition at line 249 of file implicitResidual.hh.

◆ clone()

template<class F >

virtual ImplicitResidual2D<F>* estimator::ImplicitResidual2D< F >::clone ( ) const

inlinevirtual

Implements estimator::ImplicitResidual< F >.

Definition at line 303 of file implicitResidual.hh.

◆ compute()

template<class F >

void estimator::ImplicitResidual2D< F >::compute ( )

inline

Definition at line 244 of file implicitResidual.hh.

◆ computeError_()

template<class F >

virtual void estimator::ImplicitResidual2D< F >::computeError_ ( )

inlineprivatevirtual

Definition at line 341 of file implicitResidual.hh.

◆ info()

template<class F >

virtual std::ostream& estimator::ImplicitResidual2D< F >::info ( std::ostream & os ) const

inlineprotectedvirtual

Reimplemented from estimator::ImplicitResidual< F >.

Definition at line 307 of file implicitResidual.hh.

◆ operator()()

template<class F >

virtual const Real estimator::ImplicitResidual2D< F >::operator() ( ) const

inlinevirtual

Definition at line 301 of file implicitResidual.hh.

◆ setErrorNorm()

template<class F >

void estimator::ImplicitResidual2D< F >::setErrorNorm	(	concepts::ElementFunction< F > &	func,
		const concepts::ElementFormula< F > *	w = `0`,
		Real	s = `1.0`
	)

const concepts::Vector<F>& estimator::ImplicitResidual2D< F >::sol_

private

Definition at line 339 of file implicitResidual.hh.

◆ ws_

template<class F >

concepts::Sequence<Real> estimator::ImplicitResidual2D< F >::ws_

private

Definition at line 318 of file implicitResidual.hh.

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

estimator/implicitResidual.hh

estimator::ImplicitResidual2D< F > Class Template Reference

Public Member Functions

Protected Member Functions

Protected Attributes

Private Member Functions

Private Attributes

Detailed Description

template<class F> class estimator::ImplicitResidual2D< F >

Constructor & Destructor Documentation

◆ ImplicitResidual2D()

Member Function Documentation

◆ addBoundaryData() [1/2]

◆ addBoundaryData() [2/2]

◆ addLhs()

◆ addRhs()

◆ clone()

◆ compute()

◆ computeError_()

◆ info()

◆ operator()()

◆ setErrorNorm()

Member Data Documentation

◆ a_

◆ bforms_

◆ eAttrbs_

◆ g_

◆ gAttrbs_

◆ lforms_

◆ lws_

◆ norm_d_

◆ norm_fs_

◆ norm_s_

◆ norm_ws_

◆ sol_

◆ ws_

template<class F>
class estimator::ImplicitResidual2D< F >