Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u(t) - Deltau + epsilon(-2) f (u) = 0 arising from phase transition in materials science, where - is a small parameter known as an "interaction length". The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on epsilon. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function no. In particular, all our error bounds depend on 1, only in some lower polynomial order for small epsilon. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18,19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow.