Abstractions and Automated Algorithms for Mixed Domain Finite Element Methods

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