An abstraction for solving multi-domain problems using finite element methods

TL;DR

The paper introduces a unified high-level abstraction for representing and solving multi-domain finite element problems within the mixed variational framework. It extends UFL with MeshSequence, CellSequence, and new intersection measures to express problems across conforming subdomains of potentially differing cell types, and implements these ideas in Firedrake. The authors validate the approach with quad-triangle and hex-quad mixed-domain problems and a fluid-structure interaction benchmark, demonstrating correct convergence and practical performance. The work enhances code expressiveness and productivity by enabling monolithic residual/Jacobian formulations and seamless integration with existing solver and time-stepping workflows. This abstraction lays the groundwork for tackling non-conforming and more complex multi-domain scenarios in a scalable, composable FEM workflow.

Abstract

We introduce a new abstraction for the representation and solution of multi-domain problems using finite element methods. This is an advance over previous work in that it achieves a single higher-level abstraction that represents multi-domain problems in the mixed variational problem formalism. We implemented our new abstraction in UFL and Firedrake, and validated our implementations solving a quad-triangle mixed-cell-type problem, a hex-quad mixed-cell-type problem, and a fluid-structure interaction benchmark problem.

Figures (7)

• Figure 1: An example sequence of triangular meshes $\mathcal{T}_i$ ($i=0,1,2$). Triangles in each mesh $\mathcal{T}_i$ have been drawn from the same set of triangles. On each mesh, exterior and interior facets are represented by thick and thin lines, respectively. We denote the set of cells labeled as Z as $\mathcal{T}_{\text{Z}}$, and the sets of facets labeled as A, B, C, and I as $\mathcal{F}_{\text{A}}$, $\mathcal{F}_{\text{B}}$, $\mathcal{F}_{\text{C}}$, and $\mathcal{F}_{\text{I}}$, respectively. Vertex numbers in the $\mathcal{T}_0$ numbering are shown on all meshes to make the relations between meshes clear.
• Figure 2: UFL FunctionSpace structures. \ref{['Fi:abstraction_0']}FunctionSpace abstraction. \ref{['Fi:abstraction_1']} Conventional FunctionSpace constituents for a single-domain problem. \ref{['Fi:abstraction_2']}FunctionSpace constituents for a multi-domain problem.
• Figure 3: Quad-triangle mixed cell-type problem setup.
• Figure 4: Hex-quad mixed cell-type problem setup.
• Figure 5: FSI benchmark problem setup. The meshes are at vastly reduced resolution for legibility.