Finite element approximation of power mean curvature flow

Abstract

In [21] the evolution of hypersurfaces in $\mathbb{R}^{n+1}$ with normal speed equal to a power $k>1$ of the mean curvature is considered and the levelset solution $u$ of the flow is obtained as the $C^0$-limit of a sequence $u^ε$ of smooth functions solving the regularized levelset equations. We prove a rate for this convergence. Then we triangulate the domain by using a tetraeder mesh and consider continuous finite elements, which are polynomials of degree $\le 2$ on each tetraeder of the triangulation. We show in the case $n=1$ (i.e. the evolving hypersurfaces are curves), that there are solutions $u^ε_h$ of the above regularized equations in the finite element sense, and estimate the approximation error between $u^ε_h$ and $u$. Our method can be extended to the case $n>1$, if one uses higher order finite elements.