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NA seminar: Balázs Kovács, University of Tübingen


Tid: 2020-03-09 10:30
Plats: MH:309A
Kontakt: eskil [dot] hansen [at] math [dot] lth [dot] se
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A convergent algorithm for mean curvature flow with and without forcing

Abstract We will sketch a proof of convergence is for semi- and full
discretizations of mean curvature flow of closed two-dimensional
surfaces. The proposed and studied numerical method combines evolving
surface finite elements, whose nodes determine the discrete surface
like in Dziuk's algorithm proposed in 1990, 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 (from [Huisken (1984)]) 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. We will also present various numerical experiments to
illustrate and complement the theoretical results.
Furthermore, we will give an outlook towards forced mean curvature
flow, that is, for problems coupling mean curvature flow with a surface
The talk is based on joint work with B. Li (Hong Kong) and Ch. Lubich