A convergent evolving finite element algorithm for mean curvature flow of closed surfaces
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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.
Mathematics Subject Classification35R01 65M60 65M15 65M12
We thank Frank Loose and Gerhard Wanner for helpful comments, and we also thank Jörg Nick for helpful discussions. We thank two anonymous referees for their constructive comments on a previous version. The work was partially supported by a grant from the Germany/Hong Kong Joint Research Scheme sponsored by the Research Grants Council of Hong Kong and the German Academic Exchange Service (G-PolyU502/16). The work of Balázs Kovács and Christian Lubich is supported by Deutsche Forschungsgemeinschaft, SFB 1173.
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