Numerical analysis for phase field simulations of moving interfaces with topological changes

Phase field methods are a widely accepted tool for the approximation of moving free interfaces in sharp interface problems. Topological changes in the solution, such as nucleation or vanishing of particles or merging or pinching of interfaces, lead to singularities in the free boundary. In the sharp interface model, these singularities cause both numerical and theoretical problems, whereas they are handled “automatically” in phase field simulations. Phase field models contain a length scale ε > 0 that vanishes in the sharp interface limit. Therefore, when ε → 0, practical numerical methods have to be robust in the sense that error estimates may only depend polynomially on ε^-1, not exponentially. We show that robust error control is possible past the occurrence of topological changes and without restrictive assumptions on the initial data. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)