Function space bases in the dune-functions module

TL;DR

This paper describes the design and implementation of function space bases in the dune-functions module, enabling flexible discretizations for PDEs within the Dune framework.

Contribution

It introduces an abstract, tree-based approach to function space bases, allowing diverse degree of freedom numberings and supporting complex discretizations.

Findings

Provides a comprehensive interface for function space bases

Demonstrates application with a Stokes equation solver

Enables flexible and modular discretization schemes

Abstract

The dune-functions Dune module provides interfaces for functions and function space bases. It forms one abstraction level above grids, shape functions, and linear algebra, and provides infrastructure for full discretization frameworks like dune-pdelab and dune-fem. This document describes the function space bases provided by dune-functions. These are based on an abstract description of bases for product spaces as trees of simpler bases. From this description, many different numberings of degrees of freedom by multi-indices can be derived in a natural way. We describe the abstract concepts, document the programmer interface, and give a complete example program that solves the stationary Stokes equation using Taylor-Hood elements.