Main Gallery

Discontinuous Galerkin Methods

Linear Elasticity: a deformed beam

Time Dependent Reaction Diffuison: spiral wave

Wave Equation: double slit domain (using mass lumping)

Saddle point solver for the stationary Stokes problem

Navier-Stokes: flow around a cylinder

Mixed method for the Poisson equation

Eigenvalue problems

Biharmonic problem

Adaptive phase field: crystal growth model

Moving surface grid: flow under mean curvature

Dual Weighted Residual Estimate

Extension Modules

Discontinuous Galerkin Methods with DUNE-FEM-DG

The add on module Dune-Fem-Dg provides a number of DG algorithms focusing on solving systems of evolution equations of advection-diffusion-reaction type of the form

\[\partial_t U + \nabla\cdot\Big( F_c(x,t,U) - F_v(x,t,U,\nabla U) \Big) = S(U).\]

The implemented method include a wide range of methods for DG discretization of the diffusion term including CDG2, BR2, IP, and many others. The advection term can be discretized using a local Lax-Friedrichs flux, specialized fluxes e.g. HLLE for the Euler equations, or user defined fluxes. To stabilize the DG method for advection dominated problems limiters with troubled cell indicators are available. Finally we use a method of lines approach for the time stepping based on explicit, implicit or, IMEX Runge-Kutta schemes using matrix-free Newton-Krylov solvers. As a final note the module can also be used to solve first order hyperbolic problems using a Finite-Volume method with linear reconstruction. A detailed description of the module is found in the [DK21].

Euler System of Gas Dynamics

An Advection-Diffusion-Reaction System: Chemical Reaction Problem

Virtual Element Methods with DUNE-VEM

demo notebook demo script Cahn-Hilliard notebook Cahn-Hilliard script

This module is based on dune-fem and provides implementation for the Virtual Element Method. You can install the package from Pypi by running pip install dune-vem. The sources for the methods is available in the dune-vem git repository. See our publication for details on our approach to add virtual element spaces to existing finite element software frameworks. The examples from the paper can be reproduced using the scripts collected in https://gitlab.dune-project.org/dune-fem/dune-vem-paper. For a focus on the analysis and testing of spaces for forth order problems have a look here.