WIAS Preprint No. 1416, (2009)

Optimal and robust a posteriori error estimates in $L^infty(L^2)$ for the approximation of Allen--Cahn equations past singularities



Authors

  • Bartels, Sören
  • Müller, Rüdiger
    ORCID: 0000-0003-2643-722X

2010 Mathematics Subject Classification

  • 65M60 65M15 35K55

Keywords

  • Allen-Cahn equation, mean curvature flow, finite element method, error analysis, adaptive methods

Abstract

Optimal a posteriori error estimates in $L^infty(0,T;L^2(O))$ are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.

Appeared in

  • Math. Comp., 80 (2011) pp. 761--780.

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