Concepts for Composing Finite Element Function Space Bases

TL;DR

This paper introduces software concepts for composing finite element function spaces using tree-based product space representations, enabling flexible data layouts and compatibility with various linear algebra tools.

Contribution

It presents a novel approach to handle composed function spaces with tree structures, facilitating adaptable data layouts and integration with diverse solvers.

Findings

Tree-based product space representation allows flexible degrees of freedom numbering.

The approach supports different data layouts and solver integrations.

Implementation in the dune-functions module demonstrates practical applicability.

Abstract

Finite Element discretizations of coupled multi-physics partial differential equation models require the handling of composed function spaces. In this paper we discuss software concepts and abstractions to handle the composition of function spaces, based on a representation of 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, allowing to adapt the function spaces to very different data layouts, so that it opens the possibility to directly use the finite element code with very different linear algebra codes, different data structures, and different algebraic solvers. A recurring example throughout the paper is the stationary Stokes equation with Taylor--Hood elements as these are naturally formulated as product spaces and highlight why different storage patterns are desirable. In the second half of the paper we discuss a particular realization of most of these concepts in the \dunemodule{dune-functions} module, as part of the DUNE ecosystem.