MultiMesh Finite Element Methods: Solving PDEs on Multiple Intersecting Meshes

TL;DR

This paper introduces multimesh finite element methods that allow for solving PDEs on multiple intersecting meshes, enabling efficient modeling of complex multi-body systems with large deformations.

Contribution

The paper formulates a novel multimesh finite element framework for the Poisson equation, demonstrating its robustness and optimal convergence in complex intersecting mesh scenarios.

Findings

Optimal order convergence demonstrated

Robustness against thin intersections and rounding errors shown

Applicable to large deformation multi-body problems

Abstract

We present a new framework for expressing finite element methods on multiple intersecting meshes: multimesh finite element methods. The framework enables the use of separate meshes to discretize parts of a computational domain that are naturally separate; such as the components of an engine, the domains of a multiphysics problem, or solid bodies interacting under the influence of forces from surrounding fluids or other physical fields. Such multimesh finite element methods are particularly well suited to problems in which the computational domain undergoes large deformations as a result of the relative motion of the separate components of a multi-body system. In the present paper, we formulate the multimesh finite element method for the Poisson equation. Numerical examples demonstrate the optimal order convergence, the numerical robustness of the formulation and implementation in the face of thin intersections and rounding errors, as well as the applicability of the methodology. In the accompanying paper~\cite{mmfem-2}, we analyze the proposed method and prove optimal order convergence and stability.