Detailed Description

Interface for a localized operator that fulfills the so-called \(H\)-independent DOF dependence.

A discrete operator \({\cal L}_h:{\cal V}_h \to {\cal W}_h\) fulfills the \(H\)-independent DOF dependence if any operator evaluation restricted to a single DOF \(\tau[{\cal L}_h[u_h]]\) can be computed with a constant complexity \(J\) independently of the dimension \(H\) for all DOFs \(\tau \in \Sigma({\cal W}_h)\) and functions \(u_h \in {\cal V}_h\).

Example

Grid based operators with local stencil for operator evaluations like most finite volume and finite element operators.

Definitions

Let \(\Sigma({\cal V}_h):=\{\sigma_i\}_{i=1}^H\) and \(\Sigma({\cal W}_h):=\{\tau_i\}_{i=1}^H\) be the DOFs of the discrete functions spaces \({\cal V}_h\) and \({\cal W}_h\), respectively.
We denote a map of interpolation DOF indices \({\cal J}:\{1,\ldots,M\} \to \{1,\ldots,H\}\) and
a map of restriction DOF indices \({\cal I}:\{1,\ldots,M\} \to \text{Pot}(\{1,\ldots,H\})\), such that
\[\tau_{{\cal J}(m)}\left[ {\cal L}_h \left[ \sum_{m' \in {\cal I}(m)} \sigma_{m'}[u_h] \psi_{m'} \right] \right] = \tau_{{\cal J}(m)}[{\cal L}_h[u_h]]\]
for all \(m=1,\ldots,M\) and \(u_h \in {\cal V}_h\). Here \(\psi_i \in {\cal V}_h, i=1,\ldots,H\) are basis functions of the discrete argument function space.
Sometimes it is desirable, to reduce the complexity of the discrete function spaces from \(H\) to \(M\) and \(M_{\text{ext}}\). In this case, we obtain restricted DOF sets \(\hat{\Sigma}_{M_{\text{ext}}}({\cal V}_h)\) and \(\hat{\Sigma}_{M}({\cal W}_h)\), and indices map \(\hat{\cal J}_M\) and \(\hat{\cal I}_{M_{\text{ext}}}\).

Definition at line 17 of file ILocalizedOperator.m.

Inheritance diagram for ILocalizedOperator:

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Public Member Functions
virtual function [ INC , INCoff , J ] =	apply (model, model_data, arg, NU_ind)
	compute operator evaluation on argument function for given interpolation DOF indices specified by `NU_ind` More...

virtual function ret_size =	ret_size (model_data, NU_ind)
	returns the length of the return `INC` value of an apply() method call. More...

virtual function arg_size =	arg_size (model_data, NU_ind)
	returns the length of the required argument DOF vector for an apply() method call. More...

virtual function [ nmd , eind , eind_local ] =	compute_TM_global (model_data, TM_local)
	computes a subset of restriction DOF indices \(\cup_{m \in T_{M_{\text{local}}}} \hat{{\cal I}}_{M_{\text{ext}}}(m)\) More...

virtual function ipm =	inner_product_matrix (model_data)
	return inner product matrix for range function space \({\cal Wh}\) More...

Static Public Member Functions
static function eind =	compute_TM_global_vertex (model_data, TM_local)
	Specialization of compute_TM_global() for grid based finite volume operators. More...

static function eind =	compute_TM_global_edge (model_data, TM_local, local_stencil_size)
	Specialization of compute_TM_global() for grid based finite volume operators. More...

static function opdata =	fill_opdata (rmodel,@IDetailedData detailed_data)
	This method can be called during empirical interpolation in order to collect arbitrary data (for example parameter independent or constant data) in a detailed data node. More...

static function opdata =	copy_extract_opdata (opdata_copy, rmodel, Mid)
	Used by a copy constructor of IDetailedData objects which collect operator data with fill_opdata() methods. More...

Static Public Attributes
static const	id
	a string name specifying the implemented operator More...

Member Function Documentation

function [ INC , INCoff , J ] = ILocalizedOperator.apply	(	model,
		model_data,
		arg,
		NU_ind
	)

pure virtual

compute operator evaluation on argument function for given interpolation DOF indices specified by NU_ind

Parameters

arg	the argument function \(u_h \in {\cal V}_h\)
NU_ind	interpolation DOF indices \(\nu_{\text{ind}} \subset \text{range}(\hat{\cal J}_M)\). ( Default: [] meaning \(\{1,\ldots,H\}\) )
model	model
model_data	model data

Return values

INC	vector of DOFs with indices \(\nu_{\text{ind}}\) of an operator evaluation
INCoff	optionally stores the DOFs of a constant part of an operator. This has been added for compatibility with separable affine operators.
J	Jacobian \(D[{\cal L}_h]\|_{u_h}\) restricted to the vector of interpolation DOFs. This ought to be a matrix of size \(M_{\text{ext}}\times M\) with \(M_{\text{ext}}:= \text{card}( \cup_{m\in\nu_{\text{ind}}} {\cal J}(m))\).

Implemented in LocalizedOperatorDefault.

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function arg_size = ILocalizedOperator.arg_size	(	model_data,
		NU_ind
	)

pure virtual

returns the length of the required argument DOF vector for an apply() method call.

This should usually return \(\text{card}( \cup_{m\in\nu_{\text{ind}}} {\cal J}(m))\) or \(H\).

Parameters

NU_ind	interpolation DOF indices \(\nu_{\text{ind}} \subset \text{range}(\hat{\cal J}_M)\). ( Default: [] meaning \(\{1,\ldots,H\}\) )
model_data	model data

Return values

arg_size length of the required argument DOF vector for an apply() method call.

Implemented in LocalizedOperatorDefault.

function [ nmd , eind , eind_local ] = ILocalizedOperator.compute_TM_global	(	model_data,
		TM_local
	)

pure virtual

computes a subset of restriction DOF indices \(\cup_{m \in T_{M_{\text{local}}}} \hat{{\cal I}}_{M_{\text{ext}}}(m)\)

The most common use-case is the computation of indices for grid cells in the reduced grid model_data.grid_local_ext. For most operators the neighbors of the interpolation DOFs need to be computed somehow. Default implementations for this exist as

Parameters

TM_local	a set of local DOF indices at which the operator shall be interpolated \(T_{M_{\text{local}}}\).
model_data	model data

Return values

nmd	updated `model_data` structure with new index sets \({\cal I}_{M_{\text{ext}}}\) and \({\cal J}_M\)
eind	a subset of restriction DOF indices \(\cup_{m \in T_{M_{\text{local}}}} {\cal I}(m)\)
eind_local	the new localized set of restriciton DOF indices (relative to the new `model_data` \(\cup_{m \in T_{M_{\text{local}}}} {\cal I}_{M_{\text{ext}}}(m)\)

function eind = ILocalizedOperator.compute_TM_global_edge	(	model_data,
		TM_local,
		local_stencil_size
	)

static

Specialization of compute_TM_global() for grid based finite volume operators.

If each DOF is connected to a grid cell of an underlying grid, this method selects to a set of DOFs all DOFs connected to grid cells neighboring the edges of all selected grid cells. The inclusion of edge-neighbors is repeated as often as indicated by local_stencil_size

Parameters

model_data	model data
TM_local	a set of local DOF indices at which the operator shall be interpolated \(T_{M_{\text{local}}}\).
local_stencil_size	indicates how often the inclusion of edge-neighors shall be repeated. (Default: 1)

Return values

eind	a subset of restriction DOF indices \(\cup_{m \in T_{M_{\text{local}}}} \hat{{\cal I}}_{M_{\text{ext}}}(m)\)

Required fields of model_data:

grid — a structure containing geometry information of a mesh used for discretizations

Definition at line 221 of file ILocalizedOperator.m.

function eind = ILocalizedOperator.compute_TM_global_vertex	(	model_data,
		TM_local
	)

static

Specialization of compute_TM_global() for grid based finite volume operators.

If each DOF is connected to a grid cell of an underlying grid, this method selects to a set of DOFs all DOFs connected to grid cells neighboring the vertices of all selected grid cells.

Parameters

model_data	model data
TM_local	a set of local DOF indices at which the operator shall be interpolated \(T_{M_{\text{local}}}\).

Return values

eind	a subset of restriction DOF indices \(\cup_{m \in T_{M_{\text{local}}}} \hat{{\cal I}}_{M_{\text{ext}}}(m)\)

Required fields of model_data:

grid — a structure containing geometry information of a mesh used for discretizations

Definition at line 180 of file ILocalizedOperator.m.

function opdata = ILocalizedOperator.copy_extract_opdata	(	opdata_copy,
		rmodel,
		Mid
	)

static

Used by a copy constructor of IDetailedData objects which collect operator data with fill_opdata() methods.

Parameters

opdata_copy	opdata structure to be copied
rmodel	rmodel
Mid	Id of the `detailed_data` structure, i.e. the interpolation DOFs, this operator is connected to. This can differ from the Id id.

Return values

opdata copied struct storing arbitrary data on the operator

Definition at line 271 of file ILocalizedOperator.m.

function opdata = ILocalizedOperator.fill_opdata	(		rmodel,
		@IDetailedData	detailed_data
	)

static

This method can be called during empirical interpolation in order to collect arbitrary data (for example parameter independent or constant data) in a detailed data node.

Parameters

rmodel	rmodel
detailed_data	detailed data

Return values

opdata struct storing arbitrary data on the operator

Definition at line 255 of file ILocalizedOperator.m.

function ipm = ILocalizedOperator.inner_product_matrix ( model_data )

pure virtual

return inner product matrix for range function space \({\cal Wh}\)

Parameters

model_data model data

Return values

ipm	the inner product matrix of the range function space

Implemented in LocalizedOperatorDefault.

function ret_size = ILocalizedOperator.ret_size	(	model_data,
		NU_ind
	)

pure virtual

returns the length of the return INC value of an apply() method call.

This should usually return \(\text{card}(\nu_{\text{ind}})\) or \(H\).

Parameters

NU_ind	interpolation DOF indices \(\nu_{\text{ind}} \subset \text{range}(\hat{\cal J}_M)\). ( Default: [] meaning \(\{1,\ldots,H\}\) )
model_data	model data

Return values

ret_size length of the return INC value of an apply() method call.

Implemented in LocalizedOperatorDefault.

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