Higher-order time discretizations with ALE finite elements for parabolic problems on evolving surfaces

TL;DR

This paper introduces a novel ALE finite element approach for discretizing linear evolving surface PDEs, combining spatial and temporal methods to ensure mesh regularity, stability, and optimal convergence, supported by theoretical analysis and numerical tests.

Contribution

It develops an ALE finite element method for evolving surface PDEs that maintains mesh quality and achieves stability and optimal convergence with specific time discretizations.

Findings

Unconditional stability proven for the discretization methods.

Optimal order convergence demonstrated theoretically and numerically.

Numerical experiments confirm theoretical stability and accuracy.

Abstract

A linear evolving surface partial differential equation is first discretized in space by an arbitrary Lagrangian Eulerian (ALE) evolving surface finite element method, and then in time either by a Runge-Kutta method, or by a backward difference formula. The ALE technique allows to maintain the mesh regularity during the time integration, which is not possible in the original evolving surface finite element method. Unconditional stability and optimal order convergence of the full discretizations is shown, for algebraically stable and stiffly accurate Runge-Kutta methods, and for backward differentiation formulae of order less than 6. Numerical experiments are included, supporting the theoretical results.