Error estimates for $A$-stable backward difference full discretizations of Willmore flow of closed surfaces

Abstract

A proof of optimal-order $H^1$-norm error estimates is given for $A$-stable backward difference full discretizations (of order 1 and 2) of Willmore flow for closed two-dimensional surfaces. The numerical method discretizes a coupled system of evolution equations by evolving surface finite elements of polynomial degree at least two in space and backward difference method of order 1 or 2 in time. The convergence analysis is based on a stability analysis, based on energy estimates exploiting the anti-symmetric structure of the second-order system, in combination with Dahlquist's $G$-stability and the multiplier techniques of Nevanlinna and Odeh, with a new upper bound in the spirit of Dahlquist. Numerical experiments illustrate and complement the theoretical results.