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A convergent evolving finite element algorithm for mean curvature flow of closed surfaces

Banff International Research Station for Mathematical Innovation and Discovery

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, which 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; Numerical analysis; Partial differential equations; Scientific computing

video/mp4

Author affiliation: University of Tuebingen

2019-06-06

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/70546

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/