Error Controlled Local Resolution of Evolving Interfaces for Generalized Cahn-Hilliard Equations

Sören Bartels , Rüdiger Müller

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 09-02, pp. 1-26

MSC 2000

65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

35B25 Singular perturbations

Abstract
For phase field equations of generalized Cahn-Hilliard type, we present an a posteriori error analysis that is robust with respect to a small interface length scale which enters the model as a regularizing parameter. By the solution of a fourth order elliptic eigenvalue problem in each time step we gain a fully computable error bound. In accordance with theoretical results, this error bound only depends on the inverse of the small parameter in a low order polynomial for a smooth evolution of the interface. We apply the general framework to the technologically relevant Cahn-Hilliard system coupled with homogeneous elasticity. The derived estimators can be used for adaptive mesh refinement and coarsening. In numerical examples we illustrate that the computation of the principal eigenvalue allows the detection of critical points during the time evolution like merging of interfaces or other topological changes. Moreover, it confirms theoretical predictions about fast relaxation of nonsmooth components in the initial data.