Quasi-optimal and robust a posteriori error estimates in 𝐿^{∞}(𝐿²) for the approximation of Allen-Cahn equations past singularities

Bartels, Sören; Müller, Rüdiger

doi:10.1090/S0025-5718-2010-02444-5

Quasi-optimal and robust a posteriori error estimates in $L^\infty (L^2)$ for the approximation of Allen-Cahn equations past singularities
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by Sören Bartels and Rüdiger Müller;

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

DOI: https://doi.org/10.1090/S0025-5718-2010-02444-5

Published electronically: November 18, 2010

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Abstract:

Quasi-optimal a posteriori error estimates in $L^\infty (0,T;L^2(\Omega ))$ 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.

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