Welcome to the dune-fem tutorial (for version 2.11)

This module brings python scripting support to Dune. This version describes the bindings for the development version (to become 2.11). The bindings serves three purposes:

1. High level program control for solving partial differential equations using classes from the Dune core and from Dune-Fem [BBD+08, DKNO10] with a recent update provided in [BBD+21]. The unified form language UFL [ALO+13] is used to describe the mathematical model, all realizations of the Dune grid interface can be used to work with the domain tessellation, and the finite element spaces, operator, and solvers provided by Dune-Fem for the discretizations and solving steps. All of this is available to be used in python scripts or through Jupyter notebooks.

2. Rapid prototyping of new methods or other parts of a simulation is easy since the interfaces provided are very similar to the Dune C++ interface. This makes it easy to transfer a working prototype from python (easy to develop) to C++ (high efficiency). Small C++ code snippets can be easy called from python using just in time compilation.

3. Rapid prototyping of new implementations of Dune interfaces. For example new implementations of the Dune grid interface can be easily tested. For Dune-Fem developers, new grid views, discrete function spaces, and scheme classes can be added and tested.

Table of Content

Overview

• An Overview of this Document

Installation

• From PyPi
• From Source
• Troubleshooting

Getting Started

• Introduction
  • A Laplace problem
  • Laplace equation with Dirichlet boundary conditions
  • A non-linear elliptic problem
  • Using Constant parameters and grid functions in PDEs
• General Concepts (with a time dependent problem)
  • Setting up the Grid
  • Grid Functions
  • Discrete Spaces
  • Discrete Functions
  • Operators and Schemes
  • A Nonlinear Time Dependent Problem
  • Listing Available Dune Components

Next Steps

• More General Boundary Conditions
  • Dirichlet boundary conditions
  • Accessing the Dirichlet degrees of freedom
  • Neumann and Robin boundary conditions
  • Using boundary ids (and some more complex examples)
• Tweaking the (non-) linear solvers
  • Python sided Newton solver
  • Handling Dirichlet boundary conditions
  • Other internal solvers (petsc)
  • Other internal solvers (istl)
  • Parameters for fine-tuning the internal solvers
  • Available solvers and parameters
• Alternative Solvers (Scipy and Petsc)
  • Accessing underlying data structures
  • Defining discrete function with given external dof vector
  • Using Scipy
  • Using PETSc4Py
• Examples of grid construction
  • Example using the Dune Grid Format (DGF)
  • 3D example (using PyGmsh)
  • Converting gmsh to DGF
  • Using meshio to read a gmsh file
  • 1D example
• Parallelization
  • OpenMP
  • MPI
• Backup and restore (pickling)
  • Pickling
  • A Chekpointer class
• Using the Full Grid Interface
  • Overview
  • Dense Vectors and the Geometry Classes
  • Grid Construction and Basic Interface
  • Using grid functions
  • Importing user defined C++ code
  • Attaching Data to the Grid
  • Output of Grid and Visualization of Data

Further Examples

• Discontinuous Galerkin Methods
  • Advection-Diffusion: Discontinuous Galerkin Method with Upwinding
  • A linear transport problem
• 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
  • Monolithic solver (Lagrange and DG formulation)
  • Stokes problem with fieldsplit preconditioner (PETSc)
  • (Quasi) Stokes equations (Scipy based Uzawa scheme)
• Eigenvalue problems
  • Eigenvalues of the Neumann Laplacian
  • Dirichlet Eigenvalue problem
• Biharmonic problem
  • Implementation
  • Simply supported boundary conditions
• Adaptive phase field: crystal growth model
• Moving surface grid: flow under mean curvature
• Dual Weighted Reisdual Estimate

Further Topics

• Adaptivity and Moving Domains: extended grid views
  • Overview and some basic grid views (level and filtered)
  • Dynamic Local Grid Refinement and Coarsening
  • Evolving Domains
• Using C++ Code Snippets
  • C++ Based Grid Functions
  • Sampling the Solution over a Line
  • Mean Curvature Flow (revisited)
  • Dual Weighted Reisdual Estimate (revisited)
• API for schemes and operators
  • Abstract concepts
  • Operator without DBC:
  • Operator with DBC:
  • Scheme without DBC:
  • Scheme with DBC:
  • Additional
  • Unclear

Extension Modules

• Discontinuous Galerkin Methods: the DUNE-FEM-DG Module
• Virtual Element Methods: the DUNE-VEM module

User Projects

• HP adaptive DG scheme for twophase flow problem
• Mixed-dimensional PDEs: the Dune-MMesh module

Information and Resources

• Citing this project
• Information for C++ Developers
  • Remove and rebuild existing modules
  • Using a debugger
  • List of build specific environment variables
• Contributing to this tutorial
• How to showcase your own work
• Notebooks and Scripts
• Additional Scripts
• Mesh Files used in the Examples
• Changelog
  • Since 2.9 release
  • Since 2.10 release
• Keyword index

Bibliography

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