Gerhard Dziuk, September 18, 2001

Gerhard Dziuk, University of Reiburg, Germany

"A fully discrete finite element method for anisotropicmean curvature flow"

Abstract

Anisotropic mean curvature flow is the gradient flow of weighted area of a surface. The normal velocity of the surface is given by its anisotropic mean curvature. We derive a numerical method for the resulting nonlinear differential equation

$\begin{displaymath} \begin{array}{lllll} \beta(\nabla u,-1) u_t - \sqrt{1+\ver... ...mma_{p_i p_j}(\nabla u,-1) u_{x_i x_j} & = & 0, \end{array}\end{displaymath}$

which is not in divergence form. $\gamma$ is a suitable anisotropy function and $\beta$ is a mobility. The equation describes the motion of a graph under weighted mean curvature flow, but the method can easily be used for the mean curvature flow of level sets. We discretize the problem with finite elements in space. For the time discretization we use a semi implicit scheme. The convergence result though does not require a coupling of time step size and spatial grid size, which is a quite astonishing fact.

This is joint work with K. Deckelnick (University of Sussex, Brighton).

Tuesday, September 18, 2001
Time: 1:30 P.M.
Location:Physical Plant Building Room 3008

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