A convergent evolving finite element algorithm for mean curvature flow of closed surfaces (Christian Lubich, Tuebingen)

26.03.2019 14:00

A numerical method with a proof of convergence is given for mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here uses evolving finite elements, whose nodes determine the discrete surface like in Dziuk's method from 1990. In contrast to Dziuk's approach, where a weak formulation of the formally heat-like equation describing the mean curvature flow of a closed surface is discretized and for which still no convergence result is known to date, we here discretize Huisken's evolution equations for the normal vector and mean curvature and use these evolving geometric quantities in the velocity law that is projected to the finite element space. This numerical method admits a convergence analysis, which combines stability estimates and consistency estimates to yield optimal-order (i.e., O(h^k) for finite elements of polynomial degree k \ge 2) 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. A full discretization with optimal-order error bounds is obtained with a linearly implicit backward difference formula for time integration. The talk is based on joint work with Balázs Kovács (Univ. Tübingen) and Buyang Li (Hong Kong Polytechnical Univ.).

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salle 623, Séminaire d'analyse numérique

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