(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	Fem.Lagrange.DefaultInfo
	scalar Lagrange FE functions on triangular grid More...
class	Fem.Lagrange.BubbleInfo
	scalar Lagrange FE functions (degree 1) with additional bubble function on triangular grid More...
class	Fem.BcInfo
	Class for boundary condition information. Allows more flexible usage. More...
class	Fem.CompositeFunctionSpace
	Composite function space for composition of FE function spaces on same grid. More...
class	Fem.DiscFunc
	represents a continous piecewise polynomial function of arbitrary dimension. Can be used for all finite elements (discrete function information class) implementing IFeminfo (e.g. Fem.Lagrange.DefaultInfo) More...
class	Fem.IFemInfo
	Abstract class for implementing finite elements. Fem info classes implementing this interface are compatible with Fem.DiscFunc. More...
class	Fem.OperatorsDefault
	Fem operators can be obtained from stored components. More...
class	Fv.TwoPhase.DivergenceSpace
class	Fv.TwoPhase.GlobalSpace
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	add_integral_kernels.m
	Addition of integral kernel evaluations.
file	dirichlet_dof_vector.m
	Assembly of vector enforcing the Dirichlet boundary condition.
file	eval_coefficients.m
	Concatenation of multiple parametric coefficient evaluations.
file	matrix_part.m
	Fem Matrix assembly.
file	operator.m
	Complete Fem system assembly for lin_stat and stokes models.
file	rhs_part.m
	Fem right-hand side assembly.
file	stokes_fd_nl_operator.m
	Evaluation of the Frechet-derivative of nonlinear Navier-Stokes problems for the approximation of the inf-sup constant.
file	stokes_matrix_fd_nl.m
	additional trilinear term in Frechet derivative of global operator
file	global_dof_index.m
	function computing the local-to-global dof map of a fem discrete function
file	nodes_edges_llcoord.m
	used in constructors, computes edge-local coordinates on reference triangle
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.