Convergence of finite elements on an evolving surface driven by diffusion on the surface

TL;DR

This paper analyzes the convergence of finite element methods for coupled surface evolution and diffusion PDEs, introducing a novel stability analysis that does not rely on geometric arguments, supported by numerical experiments.

Contribution

It develops a new stability and convergence analysis for evolving surface finite elements in coupled diffusion and surface evolution problems, applicable to various velocity laws.

Findings

Proved convergence of finite element discretization for evolving surfaces.

Established stability results independent of geometric complexities.

Validated theoretical results with numerical experiments.

Abstract

For a parabolic surface partial differential equation coupled to surface evolution, convergence of the spatial semidiscretization is studied in this paper. The velocity of the evolving surface is not given explicitly, but depends on the solution of the parabolic equation on the surface. Various velocity laws are considered: elliptic regularization of a direct pointwise coupling, a regularized mean curvature flow and a dynamic velocity law. A novel stability and convergence analysis for evolving surface finite elements for the coupled problem of surface diffusion and surface evolution is developed. The stability analysis works with the matrix-vector formulation of the method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments complement the theoretical results.