Error analysis for full discretizations of quasilinear parabolic problems on evolving surfaces

TL;DR

This paper establishes convergence results for fully discrete numerical methods solving quasilinear parabolic PDEs on evolving surfaces, combining spatial finite element discretization with advanced time integration schemes.

Contribution

It provides the first comprehensive convergence analysis for full discretizations of such PDEs on evolving surfaces, integrating spatial and temporal discretizations.

Findings

Optimal order error estimates for spatial discretization

Convergence results for Runge--Kutta and BDF time integrators

Stability analysis supporting full discretization convergence

Abstract

Convergence results are shown for full discretizations of quasilinear parabolic partial differential equations on evolving surfaces. As a semidiscretization in space the evolving surface finite element method is considered, using a regularity result of a generalized Ritz map, optimal order error estimates for the spatial discretization is shown. Combining this with the stability results for Runge--Kutta and BDF time integrators, we obtain convergence results for the fully discrete problems.