Maximum norm stability and error estimates for the evolving surface finite element method

TL;DR

This paper establishes maximum norm stability and error estimates for a finite element method applied to linear parabolic PDEs on evolving surfaces, ensuring accurate numerical solutions in maximum and gradient norms.

Contribution

It provides the first convergence analysis in maximum norms for finite element discretizations on evolving surfaces, including error bounds for Ritz maps and a maximum principle.

Findings

Proves convergence in $L^{inity}$ and $W^{1,inity}$ norms.

Derives error estimates for Ritz maps and their derivatives.

Establishes a weak discrete maximum principle.

Abstract

We show convergence in the natural $L^{\infty}$- and $W^{1,\infty}$-norm for a semidiscretization with linear finite elements of a linear parabolic partial differential equations on evolving surfaces. To prove this we show error estimates for a Ritz map, error estimates for the material derivative of a Ritz map and a weak discrete maximum principle.