An Eulerian Finite Element Method for PDEs in time-dependent domains

TL;DR

This paper presents a novel Eulerian finite element method for solving PDEs on moving domains, combining finite difference time discretization with a geometrically unfitted finite element approach, including stability and error analysis.

Contribution

It introduces a new Eulerian finite element method for PDEs in evolving domains, with comprehensive stability and error analysis, and demonstrates its efficiency through numerical examples.

Findings

Method achieves stability for PDEs on moving domains.

Error analysis accounts for discretization and geometric errors.

Numerical examples confirm practical efficiency.

Abstract

The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain. The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. This paper includes a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. Several numerical examples illustrate the theory and demonstrate the practical efficiency of the method.