Evolving finite elements for advection diffusion with an evolving interface

TL;DR

This paper develops an evolving finite element scheme for parabolic interface problems, providing optimal error bounds and numerical verification of convergence orders.

Contribution

It introduces a weak formulation and finite element method tailored for evolving interfaces in parabolic equations, with proven optimal error bounds.

Findings

Optimal order error bounds are established for arbitrary order evolving finite elements.

Numerical results verify the theoretical convergence rates.

The scheme effectively approximates problems with moving interfaces.

Abstract

The aim of this paper is to develop a numerical scheme to approximate evolving interface problems for parabolic equations based on the abstract evolving finite element framework proposed in (C M Elliott, T Ranner, IMA J Num Anal, 41:3, 2021, doi:10.1093/imanum/draa062). An appropriate weak formulation of the problem is derived for the use of evolving finite elements designed to accommodate a moving interface. Optimal order error bounds are proved for arbitrary order evolving isoparametric finite elements. The paper concludes with numerical results for a model problem verifying orders of convergence.