(M1) Discretization methods

Detailed Description

Implementation of grid-based discretization methods for parametrized partial differential equations.

This module provides

• rectangular and triangular grids for the discretization of domain spaces \( \Omega \subset \mathbb{R}^d \),
• several discretization operators acting on grid based discrete functions, and
• parametrized analytical data functions with decomposed implementations as a linear combinations of products of parameter independent and space independent terms, if possible.

In order to be compliant with the reduced basis framework described in Module (M3), a numerical scheme implemented with the routines above, should be wrapped in an IDetailedModel implementation.

Example usage of a detailed model implementation

In this section we want to show the steps that need to be undertaken in order to execute a detailed simulation for a convection diffusion equation problem.

The ModelDescr struct for this problem is returned by convdiff_model(). Starting from this description, we can construct a LinEvol.DetailedModel instance by:

model_descr = convdiff_model();
dmodel      = gen_detailed_model(model_descr);

We check the type of this model

disp(['Detailed model is of type: ', class(dmodel)]);

and find it to be a LinEvol.DetailedModel.

Now, we can compute the ModelData structure and compute a single detailed simulation snapshot:

model_data = gen_model_data(dmodel);
set_mu(dmodel, mu);
sim_data   = detailed_simulation(dmodel, model_data);

Further ready-to-use description structs can be found in the models/ directory.

Collaboration diagram for (M1) Discretization methods:

Classes
class	ModelData
	Struct with high dimensional data needed for a high dimensional simulation. More...
class	ModelDescr
	Struct with control fields for the analytical PDE functions, the discretization and the parametrization. More...
class	Test.DetailedModel
	a test model doing nothing More...
class	Fv.TwoPhase.DivergenceSpace
class	Fv.TwoPhase.PressureMean
class	Fv.TwoPhase.SaturationSpace
class	Fv.TwoPhase.VelocitySpace
class	ILocalizedOperator
	Interface for a localized operator that fulfills the so-called \(H\)-independent DOF dependence. More...
class	ISeparableFunction
class	ISeparableOperator
class	LocalizedOperatorDefault
class	SeparableFunctionDefault

Modules
	Grid generation
	Classes for the discretization of a domain space \( \Omega \subset \mathbb{R}^d \) by grids.
	Data functions
	Collection of (affinely decomposable) analytical data functions \( \mu \mapsto f:\mathbb{R} \to \mathbb{R}\).
	Finite volume
	Collection of discretized finite volume operators.
	Local discontinuous Galerkin
	Local basis functions for local discontinuous Galerin (LDG) discretizations and a collection of discretized operators acting on an LDG space.
	Finite element
	Local basis functions for finite element (FEM) discretizations and a collection of discretized operators acting on a FEM function space.

Files
file	copy_model_data_to_plot_params.m
	Helper function copying extracting relevant information for plot_params from the model.
file	plot_sequence.m
	plotting a sequence of data slices on polygonal 2d grid (constructed from params if empty) and providing a slider for changing between the data slices.
file	plot_subplot_sequence.m
	plotting a sequence of data slices on polygonal 2d grid (constructed from params if empty) and providing a slider for changing between the data slices.