Abstractions and automated algorithms for mixed domain finite element methods

TL;DR

This paper develops general abstractions and algorithms for finite element discretizations of mixed domain and mixed dimensional PDEs, enabling efficient solutions of complex coupled systems across various scientific fields.

Contribution

It introduces high-level mathematical software abstractions and algorithms for mixed domain and mixed dimensional PDEs, implemented in FEniCS, with practical examples.

Findings

Successfully implemented in FEniCS

Handled co-dimension up to one

Demonstrated on diverse scientific problems

Abstract

Mixed dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such equations naturally arise in a wide range of scientific fields including geology, physiology, biology and fracture mechanics. Mixed dimensional PDEs are also commonly encountered when imposing non-standard conditions over a subspace of lower dimension e.g. through a Lagrange multiplier. In this paper, we present general abstractions and algorithms for finite element discretizations of mixed domain and mixed dimensional PDEs of co-dimension up to one (i.e. nD-mD with |n-m| <= 1). We introduce high level mathematical software abstractions together with lower level algorithms for expressing and efficiently solving such coupled systems. The concepts introduced here have also been implemented in the context of the FEniCS finite element software. We illustrate the new features through a range of examples, including a constrained Poisson problem, a set of Stokes-type flow models and a model for ionic electrodiffusion.