Concepts for Composing Finite Element Function Space Bases

2025-08-13

AI Summary

Modeling coupled multiphysics PDEs via finite element discretization faces challenges in representing composite function spaces, rigid degree-of-freedom (DoF) numbering schemes, and poor interoperability with linear algebra backends. Method: This paper introduces a tree-based abstraction for function spaces, wherein product spaces are represented hierarchically as trees; a unified multi-index mechanism generates diverse DoF numbering strategies—supporting heterogeneous data layouts such as block-structured and interleaved formats. Contribution/Results: Implemented in the dune-functions module of the DUNE framework, the approach significantly enhances interoperability with algebraic solvers, sparse matrix formats, and parallel data structures. Experiments demonstrate efficient modeling of canonical multiphysics problems—including Taylor–Hood discretizations of the Stokes equations—while maintaining scalability and flexibility across diverse solvers (e.g., UMFPACK, PETSc) and storage layouts.

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• Machine Learning: Matrix & Tensor Methods
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• Search and Optimization: Mixed Discrete/Continuous Search

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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.

Problem

• Composing finite element function spaces for coupled multi-physics models
• Deriving multi-index degree-of-freedom numbering from product space trees
• Enabling compatibility with diverse linear algebra codes and solvers

Innovation

• Tree-based representation of product spaces
• Multi-index numbering for degrees of freedom
• Compatible with various linear algebra implementations