High-order evolving surface finite element method for parabolic problems on evolving surfaces

TL;DR

This paper develops and proves the convergence of high-order finite element methods for solving parabolic PDEs on evolving surfaces, including spatial and full discretisations with advanced time-stepping schemes.

Contribution

It introduces high-order spatial discretisations for evolving surface PDEs and provides rigorous convergence analysis for both spatial and full discretisations using modern time-stepping methods.

Findings

Proved convergence of high-order evolving surface finite element method.

Established error estimates for geometric approximation and perturbation.

Demonstrated convergence of full discretisations with backward difference and Runge-Kutta methods.

Abstract

High-order spatial discretisations and full discretisations of parabolic partial differential equations on evolving surfaces are studied. We prove convergence of the high-order evolving surface finite element method, by showing high-order versions of geometric approximation errors and perturbation error estimates and by the careful error analysis of a modified Ritz map. Furthermore, convergence of full discretisations using backward difference formulae and implicit Runge-Kutta methods are also shown.