Linearly implicit full discretization of surface evolution

TL;DR

This paper introduces a stable, higher-order full discretization method for surface evolution equations, combining ESFEM for space and linearly implicit BDF for time, with proven stability and demonstrated convergence.

Contribution

It presents a novel higher-order, linearly implicit discretization scheme for surface evolution equations, with rigorous stability analysis and numerical validation.

Findings

The discretization scheme is stable under certain conditions.

Numerical examples confirm the convergence rates.

The geometry influences consistency errors but not stability.

Abstract

Stability and convergence of full discretizations of various surface evolution equations are studied in this paper. The proposed discretization combines a higher-order evolving-surface finite element method (ESFEM) for space discretization with higher-order linearly implicit backward difference formulae (BDF) for time discretization. The stability of the full discretization is studied in the matrix--vector formulation of the numerical method. The geometry of the problem enters into the bounds of the consistency errors, but does not enter into the proof of stability. Numerical examples illustrate the convergence behaviour of the full discretization.