An Overview of this Document

This document tries to describe the main concepts needed to get a new user started on solving complex partial differential equations using the Dune-Fem python bindings. Some parts are still quite raw and we would greatly appreciate any help improving this document. In addition if you would like to promote your own work then please upload a script showcasing some simulation you did based on this package - it is possibly as easy as providing us with a Python script. More details on how to help us improve and extend this document see the section on Contributing to this tutorial.

The Python bindings are based on pybind11 [WRM17] and a detailed description on how we export the polymorphic interfaces in Dune is provided in [DN18] which describes the Dune module Dune-Python which after the Dune 2.7 release has been directly integrated into the Dune core modules. If you have a version newer than 2.8 then note that the Dune-Python module is obsolete.

Here is a quick summary of the different parts of this document - all of the code is available for download in form of both Notebooks and Scripts. Or can be obtained by cloning the git repository https://gitlab.dune-project.org/dune-fem/dune-fempy where the scripts can be found in the demo folder.

1. First off some remarks on getting the package to work: the simplest approach is to use the Python package index (pip) to install the software into a new virtual environment. Working from the git sources is a good option for more advanced users who want to change the underlying Dune C++ source files.

2. A scalar Laplace problem and a simple scalar, non linear time dependent partial differential equation are used to describe the basic concepts. This leads through the steps required to set up the problem, solve the system of equations, and visualize the results.

3. After the introduction to the underlying concepts and how to get a simple problem up and running, we discuss:

   • how to add more complicated boundary conditions to the problem formulation.

   • how to use different solver backends (including build in solvers, solvers and preconditioners available in Dune-Istl, scipy, PETSc and also petsc4py see also [BAA+18, JOP+--]).

   • how to enable either multithreading or MPI parallelization for your problem

   • how to backup and restore data including discrete functions and the grid hierarchy as is useful for example to checkpoint a simulation.

   • more details on constructing a grid using different formats including for example the Dune Grid Format (DGF) or gmsh but also using simple python structures like dictionaries for describing general unstructured grids is available. Results can be plotted using matplotlib (for 2d) and for example mayavi (for 2d and 3d). For more advanced plotting options data can be exported using vtk format and then analysed for example using paraview. All of this is demonstrated in the provided examples.

   • discuss the grid interface in more details, i.e., how to iterate over the grid, access geometrical information, and attach data to elements of the grid.

4. Building upon the general concepts described in the first part, we provide scripts showing how to solve far more complex PDEs. The Notebooks and Scripts used for each of these and the following examples can be downloaded and are hopefully useful as starting point for new projects.

5. We then discuss some further topics:

   • Local grid refinement and coarsening is a central feature of Dune-Fem. Here we show how to use it for stationary and time dependent problems. Grid adaptivity makes use of special grid views.

   • Other views are also available, one of these can be used to deform the grid given an arbitrary (discrete) vector field. This is used to compute the evolution of a surface under mean curvature flow.

   • We complete our discussion by demonstrating how to straightforwardly extend the functionality of the package and improve performance by important additional C++ algorithms and classes.

6. There are two larger extension modules build on top of dune-fem which provide Python bindings: those are the Dune-Fem-Dg and the Dune-Vem modules which focus on Discontinuous Galerkin methods for advection-diffusion (especially advection dominated problems) and the implementations of the virtual element method, respectively.

7. Finally other projects are presented some of them developed by the authors of this document, some contributed by other users. If you have used this package then we would like to hear about it and would ask you to contribute to this chapter. Doing so is quite easy (see Contributing to this tutorial for details).