Mathematical and computational aspects of solving mixed-domain problems using the finite element method

Abstract

This work focuses on mixed-domain problems, where fields defined over different domains are governed by different partial differential equations and may be coupled. Multiphysics problems, where different regions are governed by different (interacting) physical laws, are examples of mixed-domain problems. Mixed-domain problems also arise from the need to employ stable and accurate discretisations tailored to the mathematical nature of the governing equations in each domain. This work develops a new finite element framework for solving mixed-domain problems in FEniCSx that uses efficient, parallel, and scalable algorithms. The framework supports an arbitrary number of domains of possibly different topological dimensions, a range of arbitrarily high-order finite elements, several cell types, and high-order geometry. We demonstrate how solvers for a range of applications can be implemented, including Lagrange multiplier problems, domain decomposition methods, hybridised discontinuous Galerkin methods, and multiphysics problems. Performance results show that the algorithms scale well to thousands of processes. In addition, a hybridised discontinuous Galerkin (HDG) method for the incompressible Stokes and Navier–Stokes equations is generalised to a range of cell types, focusing on preserving a key invariance property of the continuous problem. This invariance property states that any irrotational component of the prescribed force is exactly balanced by the pressure gradient and does not influence the velocity field, and it can be preserved in the discrete problem if the incompressibility constraint is satisfied in a sufficiently strong sense. We derive sufficient conditions to guarantee discretely divergence-free functions are exactly divergence-free and give examples of divergence-free finite elements on meshes containing triangular, quadrilateral, tetrahedral, or hexahedral cells generated by a (possibly non-affine) map from their respective reference cells. We also prove an optimal, pressure-robust error estimate for quadrilateral cells that does not depend on the pressure approximation. The scheme is implemented using the mixed-domain framework and numerical results are provided to support our theoretical analysis. The numerical results also suggest that pressure robustness is preserved on curved cells and that high aspect ratio tensor product cells can be used in boundary layers. A scalable solver is implemented for the statically condensed Stokes system, and performance results show that the scheme is suitable for large-scale problems. A range of multiphysics problems are also considered. We use the mixed-domain framework to employ stable and accurate discretisations in each domain and demonstrate how both monolithic and partitioned coupling schemes can be implemented in a flexible manner. One example focuses on solving the magnetohydrodynamics equations in a domain with both solid and fluid regions. We use a fully coupled (monolithic) scheme that conserves mass exactly and yields a magnetic induction that is exactly solenoidal. Performance results show that the mixed-domain algorithms scale well in parallel.