Numerical approximation of level set power mean curvature flow

Abstract : In this paper we investigate the numerical approximation of a variant of the mean curvature flow. We consider the evolution of hypersurfaces with normal speed given by $H^k$, $k \ge 1$, where $H$ denotes the mean curvature. We use a level set formulation of this flow and discretize the regularized level set equation with fi nite elements. In a previous paper we proved an a priori estimate for the approximation error between the finite element solution and the solution of the original level set equation. We obtained an upper bound for this error which is polynomial in the discretization parameter and the reciprocal regularization parameter. The aim of the present paper is the numerical study of the behavior of the evolution and the numerical veri cation of certain convergence rates. We restrict the consideration to the case that the level set function depends on two variables, i.e. the moving hypersurfaces are curves. Furthermore, we con firm for specifi c initial curves and di fferent values of $k$ that the flow improves the isoperimetrical defi cit.
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Axel Kröner, Eva Kröner, Heiko Kröner. Numerical approximation of level set power mean curvature flow. [Research Report] INRIA Saclay. 2015. ⟨hal-01138347v1⟩

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