A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality
Department of Mathematics, University of Maryland, College Park, MD 20742, USA. firstname.lastname@example.org.
2 Institute for Physical Sciences and Technology, College Park, MD 20742, USA.
3 Zentrum für Technomathematik, Universität Bremen, Bibliothekstrasse 1, 28359 Bremen, Germany.
Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter ε that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε-2. Using an energy argument combined with a topological continuation argument and a spectral estimate, we establish an a posteriori error control result with only a low order polynomial dependence in ε-1. Our result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are derived to illustrate the theory.
Mathematics Subject Classification: 65M15 / 65M50 / 65M60
Key words: A posteriori error estimates / phase-field models / adaptive finite element method.
© EDP Sciences, SMAI, 2004