FINITE ELEMENT APPROXIMATION OF LEVEL SET MOTION BY POWERS OF THE MEAN CURVATURE

In this paper we study the level set formulations of certain geometric evolution equations from a numerical point of view. Specifically, we consider the flow by powers greater than one of the mean curvature (PMCF) and the inverse mean curvature flow (IMCF). Since the corresponding equations in level set form are quasi-linear, degenerate, and especially possibly singular a regularization method is used in the literature to approximate these equations to overcome the singularities of the equations. The regularized equations depend both on an regularization parameter and in case of the ICMF additionally on further parameters. Motivated by Feng, Neilan, and Prohl [Numer. Math., 108 (2007), pp. 93-119], who study the finite element approximation of IMCF, we prove error estimates for the finite element approximation of the regularized equations for PMCF. We validate the rates with numerical examples. Additionally, the regularization error in the rotationally symmetric case for both flows is numerically analyzed in a two-dimensional setting. Therefore, in the case of IMCF we fix the additional parameters. Furthermore, having the goal to estimate the regularization error we derive barriers for the regularized level set IMCF respecting all parameters and specify them further in a rotationally symmetric simplified case. At the end of the paper we present simulations in the three-dimensional case.