ANALYSIS OF CONSTANTS IN ERROR ESTIMATES FOR THE FINITE ELEMENT APPROXIMATION OF REGULARIZED NONLINEAR GEOMETRIC EVOLUTION EQUATIONS

摘要

For degenerate elliptic and possibly singular geometric evolution equations such as the level set formulations for the inverse mean curvature flow and the flow by (powers of the) mean curvature, a common procedure to overcome the possible singularity of the equation is elliptic regularization. This procedure generates regularized equations containing a regularization parameter epsilon which are by nature different from the original equations but have turned out to be a useful starting point for the proof of existence of solutions of the original equations as well as for a finite element approximation of the original equations. This paper is devoted to a first theoretical study of the dependence of constants on epsilon which appear in error estimates in the case of the regularized level set flow by powers of the mean curvature and the regularized level set inverse mean curvature flow. The obtained relation holds for both equations and is exponential in inverse powers of the regularization parameter. We work out the rather implicit relation of constants on the regularization parameter explicitly but at the price that the order of the finite elements needed is three when the space dimension of the ambient space is three. Having established such an explicit relation one can obtain a full error estimate by combining this with an estimate of the regularization error which is usually a purely analytical issue and is not considered in our present paper.