This paper is dedicated to Gerhard Dziuk on the occasion of his 70th birthday and to Gerhard Huisken on the occasion of his 60th birthday.Abstract A proof of convergence is given for semi-and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the discrete surface like in Dziuk's method, and linearly implicit backward difference formulae for time integration. The proposed method differs from Dziuk's approach in that it discretizes Huisken's evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. The error analysis combines stability estimates and consistency estimates to yield optimal-order H 1 -norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix-vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.Keywords mean curvature flow · geometric evolution equations · evolving surface finite elements · linearly implicit backward difference formula · stability · convergence analysis Mathematics Subject Classification (2000) 35R01 · 65M60 · 65M15 · 65M12

Space and time discretisations of parabolic differential equations with dynamic boundary conditions are studied in a weak formulation that fits into the standard abstract formulation of parabolic problems, just that the usualThe class of parabolic equations considered includes linear problems with time-and space-dependent coefficients and semi-linear problems such as reaction-diffusion on a surface coupled to diffusion in the bulk. The spatial discretisation by finite elements is studied in the proposed framework, with particular attention to the error analysis of the Ritz map for the elliptic bilinear form in relation to the inner product, both of which contain boundary integrals. The error analysis is done for both polygonal and smooth domains. We further consider mass lumping, which enables us to use exponential integrators and bulk-surface splitting for time integration.

Abstract. It is shown that for a parabolic problem with maximal L p -regularity (for 1 < p < ∞), the time discretization by a linear multistep method or Runge-Kutta method has maximal ℓ p -regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank-Nicolson method, the second-order backward difference formula (BDF), and the Radau IIA and Gauss Runge-Kutta methods of all orders preserve maximal regularity. The proof uses Weis' characterization of maximal L p -regularity in terms of R-boundedness of the resolvent, a discrete operator-valued Fourier multiplier theorem by Blunck, and generating function techniques that have been familiar in the stability analysis of time discretization methods since the work of Dahlquist. The A(α)-stable higher-order BDF methods have maximal ℓ p -regularity under an R-boundedness condition in a larger sector. As an illustration of the use of maximal regularity in the error analysis of discretized nonlinear parabolic equations, it is shown how error bounds are obtained without using any growth condition on the nonlinearity or for nonlinearities having singularities.

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.

High-order spatial discretisations and full discretisations of parabolic partial differential equations on evolving surfaces are studied. We prove convergence of the high-order evolving surface finite element method, by showing high-order versions of geometric approximation errors and perturbation error estimates and by the careful error analysis of a modified Ritz map. Furthermore, convergence of full discretisations using backward difference formulae and implicit Runge-Kutta methods are also shown.