L²-estimates for the evolving surface finite element method

In this paper we consider the evolving surface finite element meth- od for the advection and diffusion of a conserved scalar quantity on a moving surface. In an earlier paper using a suitable variational formulation in time dependent Sobolev space we proposed and analysed a finite element method using surface finite elements on evolving triangulated surfaces. An optimal order H¹-error bound was proved for linear finite elements. In this work we prove the optimal error bound in L²(Γ (t)) uniformly in time.