Maximal regularity of evolving FEMs for parabolic equations on an evolving surface

TL;DR

This paper proves that evolving finite element methods for parabolic equations on moving surfaces maintain maximal L^p-regularity at the discrete level, extending stationary surface results to evolving surfaces using advanced analytical techniques.

Contribution

It introduces a novel proof that extends maximal regularity results from stationary to evolving surfaces for finite element discretizations of parabolic equations.

Findings

Maximal L^p-regularity is preserved in evolving surface FEMs.

Results are first established on stationary surfaces and then extended to evolving surfaces.

The approach combines Green's functions, local energy estimates, and perturbation arguments.

Abstract

In this paper, we prove that spatially semi-discrete evolving finite element method for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^p$-regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving finite element method, properties of Green's functions on (discretised) closed surfaces, and local energy estimates for finite element methods