Finite element analysis for a diffusion equation on a harmonically evolving domain

TL;DR

This paper analyzes the convergence of finite element methods applied to a diffusion equation on a domain that evolves over time, with a focus on stability and numerical validation.

Contribution

It introduces a stability analysis framework that avoids geometric arguments and demonstrates convergence for evolving domains using numerical experiments.

Findings

Proven convergence rates for the evolving finite element semi-discretization.

Stability analysis based on matrix-vector formulation without geometric assumptions.

Numerical experiments confirming theoretical convergence results.

Abstract

We study convergence of the evolving finite element semi-discretization of a parabolic partial differential equation on an evolving bulk domain. The boundary of the domain evolves with a given velocity, which is then extended to the bulk by solving a Poisson equation. The numerical solution to the parabolic equation depends on the numerical evolution of the bulk, which yields the time-dependent mesh for the finite element method. The stability analysis works with the matrix-vector formulation of the semi-discretization only and does not require geometric arguments, which are then required in the proof of consistency estimates. We present various numerical experiments that illustrate the proven convergence rates.