A Fully Discrete Evolving Surface Finite Element Method

Abstract

In this paper we consider a time discrete evolving surface finite element method for the advection and diffusion of a conserved scalar quantity on a moving surface. In earlier papers using a suitable variational formulation in time dependent Sobolev space we proposed and analyzed a finite element method using surface finite elements on evolving triangulated surfaces [IMA J. Numer Anal., 25 (2007), pp. 385--407; Math. Comp., to appear]. Optimal order $L^2(\Gamma(t))$ and $H^1(\Gamma(t))$ error bounds were proved for linear finite elements. In this work we prove optimal order error bounds for a backward Euler time discretization.