Analysis of Highly Accurate Finite Element Based Algorithms for Computing Distances to Level Sets

The signed distance function d to an embedded (hyper-)surface Γ is required in the analysis and implementation of some higher-order methods for the numerical treatment of partial differential equations on surfaces. Two algorithms for the approximation of d are presented in this paper which require only a finite element approximation of a (smooth) level set function of Γ. One method is based on a semismooth Newton method; the other method is a nested fixed point iteration. Both are generalizations of known methods. We provide full (local) convergence analyses. Moreover, the methods are compared in two numerical experiments.