Convergence of a fully discrete approximation for advected mean curvature flows
Thomas R. Kuhn · IMA Journal of Numerical Analysis · 1998
Let Γ (t) be a closed curve in R2 which propagates in its normal direction n with velocity V = -κ-q.n-g, where κ is the mean curvature of Γ(t) and g and q are given represent, respectively, a forcing term and a vector field. In this paper we prove that such flows can be approximated by numerical solutions of advection Allen-Cahn equations. It is shown that the zero level set of the fully discrete solution using explicit time stepping converges even past singularities to the true interface provided that no fattening occurs and τ, h2 ≈ O(ε4), where h and τ denote the mesh size and the time step. For smooth flows an optimal O(ε2)-rate of convergence is derived provided τ, h2 ≈ O(ε5). The analysis is based on constructing fully discrete barriers via an explicit parabolic projection and Lipschitz dependence of the viscosity solutions with respect to perturbations of data.