Error analysis of a finite element scheme for parametric mean curvature flow based on the DeTurck trick

TL;DR

This paper analyzes the error of a finite element scheme for parametric mean curvature flow using the DeTurck trick, proving optimal error estimates and confirming them through numerical experiments.

Contribution

It provides the first rigorous error analysis for this scheme, establishing optimal $H^1$-error bounds for finite element discretizations of order $k \\geq 2$.

Findings

Optimal $H^1$-error estimates are proven for the scheme.

Numerical experiments confirm the theoretical error bounds.

The scheme maintains good mesh point distribution properties.

Abstract

The paper is concerned with the error analysis of a numerical scheme for the approximation of parametric mean curvature flow. The scheme we study is based on a reparametrization using the DeTurck trick and was proposed by Elliott and Fritz in [15]. In the semidiscrete case, for a spatial discretization by finite elements of order $k \geq 2$ we prove an optimal $H^1$-error estimate for the position vector. We present numerical experiments that confirm this error bound and demonstrate that the scheme has good properties with respect to the distribution of mesh points as already observed in [15].