Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo > Class Template Reference

Defines assembly of hierarchically extended problems for defining DLY style error estimators. More...

#include <hierarchicErrorEstimator.hh>

Detailed Description

template<class LinearFunctional, class ExtensionAnsatzVariables, class ExtensionTestVariables = ExtensionAnsatzVariables, class D2BlockInfo = HierarchicErrorEstimatorDetail::DefaultD2<LinearFunctional>>
class Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo >

Defines assembly of hierarchically extended problems for defining DLY style error estimators.

This class defines the weak formulation of hierarchical error estimation. It is a model of the LinearVariationalProblem concept.

Assume you have discretized a linear(ized) variational functional using a Galerkin ansatz space \( V_l \). The Hessian is \( A_{ll} \) and the gradient at \(x_0=0 \) is \( b_l \), such that the solution (minimizer) \( x_l \) satisfies \( A_{ll}x_l = -b_l \).

Now we extend the ansatz space by another (probably higher order) ansatz space \( V_h \). The complete system to be solved would be

\[ \begin{bmatrix} A_{ll} & A_{lh} \\ A_{hl} & A_{hh} \end{bmatrix} \begin{bmatrix} x_l \\ x_h \end{bmatrix} = - \begin{bmatrix} b_l \\ b_h \end{bmatrix}. \]

Since the assembly and solution of the large system is quite expensive, we fix \( x_l \) obtained before and solve the defect equation

\[ A_{hh} x_h = -(b_h+A_{hl}x_l) \]

instead. And since the error \( x_h \) is structurally oscillatory, one Jacobi iteration will do for error estimation purposes, such that we only need to assemble the diagonal of \( A_{hh} \) (lumping).

Important things to note:

• The assembly produces \( A_{hh} \) as matrix and \( b_h+A_{hl}x_l \) as right hand side. Make sure to provide \( x_l \) with the correct sign!
• During the assembly, both original and extension ansatz functions need to be evaluated. Remember to provide all(!) required spaces. The easiest method to do so is to extend the space list of the original problem by the extension spaces, and to shift the space index of the extension variables accordingly.

Template Parameters

LinearFunctional	the linear variational weak formulation for which the error will be estimated
ExtensionAnsatzVariables	a description of the ansatz variables and hierarchic extension spaces
ExtensionTestVariables	a description of the test variables and hierarchic extension spaces
D2BlockInfo	proivdes static block information

D2BlockInfo defines which blocks should be assembled and which ones should be lumped (only diagonal assembled). The default is HierarchicErrorEstimatorDetail::DefaultD2, which takes only the symmetric blocks and lumps all of them.

Definition at line 152 of file fem/hierarchicErrorEstimator.hh.

Public Types
typedef LinearFunctional::Scalar	Scalar
typedef ExtensionAnsatzVariables	AnsatzVars
typedef ExtensionTestVariables	TestVars
typedef FEAnsatzVars::VariableSet	FESolution

Public Member Functions
	HierarchicErrorEstimator (LinearFunctional const &lf_, FESolution const &ul_)
DomainCache	createDomainCache (int flags) const
BoundaryCache	createBoundaryCache (int flags) const
int	integrationOrder (Cell const &cell, int shapeFunctionOrder, bool boundary) const

Static Public Attributes
static ProblemType const	type = LinearFunctional::type

Member Typedef Documentation

AnsatzVars

template<class LinearFunctional , class ExtensionAnsatzVariables , class ExtensionTestVariables = ExtensionAnsatzVariables, class D2BlockInfo = HierarchicErrorEstimatorDetail::DefaultD2<LinearFunctional>>

typedef ExtensionAnsatzVariables Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo >::AnsatzVars

Definition at line 162 of file fem/hierarchicErrorEstimator.hh.

FESolution

template<class LinearFunctional , class ExtensionAnsatzVariables , class ExtensionTestVariables = ExtensionAnsatzVariables, class D2BlockInfo = HierarchicErrorEstimatorDetail::DefaultD2<LinearFunctional>>

typedef FEAnsatzVars::VariableSet Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo >::FESolution

Definition at line 165 of file fem/hierarchicErrorEstimator.hh.

Scalar

template<class LinearFunctional , class ExtensionAnsatzVariables , class ExtensionTestVariables = ExtensionAnsatzVariables, class D2BlockInfo = HierarchicErrorEstimatorDetail::DefaultD2<LinearFunctional>>

typedef LinearFunctional::Scalar Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo >::Scalar

Definition at line 161 of file fem/hierarchicErrorEstimator.hh.

TestVars

template<class LinearFunctional , class ExtensionAnsatzVariables , class ExtensionTestVariables = ExtensionAnsatzVariables, class D2BlockInfo = HierarchicErrorEstimatorDetail::DefaultD2<LinearFunctional>>

typedef ExtensionTestVariables Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo >::TestVars

Definition at line 163 of file fem/hierarchicErrorEstimator.hh.

Constructor & Destructor Documentation

HierarchicErrorEstimator()

template<class LinearFunctional , class ExtensionAnsatzVariables , class ExtensionTestVariables = ExtensionAnsatzVariables, class D2BlockInfo = HierarchicErrorEstimatorDetail::DefaultD2<LinearFunctional>>

Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo >::HierarchicErrorEstimator ( LinearFunctional const &  lf_, FESolution const &  ul_  )

inline

Parameters

lf_	the linear weak formulation for which the error will be estimated. This object can be temporary, since a copy is held internally. Thus, it should not be too large.
ul_	the approximate solution of lf_ on the lower order ansatz space.

Definition at line 437 of file fem/hierarchicErrorEstimator.hh.

Member Function Documentation

createBoundaryCache()

template<class LinearFunctional , class ExtensionAnsatzVariables , class ExtensionTestVariables = ExtensionAnsatzVariables, class D2BlockInfo = HierarchicErrorEstimatorDetail::DefaultD2<LinearFunctional>>

BoundaryCache Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo >::createBoundaryCache ( int  flags ) const

inline

Definition at line 445 of file fem/hierarchicErrorEstimator.hh.

createDomainCache()

template<class LinearFunctional , class ExtensionAnsatzVariables , class ExtensionTestVariables = ExtensionAnsatzVariables, class D2BlockInfo = HierarchicErrorEstimatorDetail::DefaultD2<LinearFunctional>>

DomainCache Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo >::createDomainCache ( int  flags ) const

inline

Definition at line 440 of file fem/hierarchicErrorEstimator.hh.

integrationOrder()

template<class LinearFunctional , class ExtensionAnsatzVariables , class ExtensionTestVariables = ExtensionAnsatzVariables, class D2BlockInfo = HierarchicErrorEstimatorDetail::DefaultD2<LinearFunctional>>

int Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo >::integrationOrder ( Cell const &  cell, int  shapeFunctionOrder, bool  boundary  ) const

inline

Definition at line 450 of file fem/hierarchicErrorEstimator.hh.

Member Data Documentation

type

template<class LinearFunctional , class ExtensionAnsatzVariables , class ExtensionTestVariables = ExtensionAnsatzVariables, class D2BlockInfo = HierarchicErrorEstimatorDetail::DefaultD2<LinearFunctional>>

ProblemType const Kaskade::HierarchicErrorEstimator< LinearFunctional, ExtensionAnsatzVariables, ExtensionTestVariables, D2BlockInfo >::type = LinearFunctional::type

static

Definition at line 160 of file fem/hierarchicErrorEstimator.hh.

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

• fem/hierarchicErrorEstimator.hh