A Posteriori Error Estimates and an Adaptive Finite Element Method for the Allen--Cahn Equation and the Mean Curvature Flow

Authors:
Xiaobing Feng;Hai-Jun Wu
Affiliations:
Department of Mathematics, University of Tennessee, Knoxville, USA 37996;Department of Mathematics, Nanjing University, Jiangsu Province, PR China
Venue:
Journal of Scientific Computing
Year:
2005

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Abstract

This paper develops an a posteriori error estimate of residual type for finite element approximations of the Allen--Cahn equation ut 驴驴u+ 驴驴2 f(u)=0. It is shown that the error depends on 驴驴1 only in some low polynomial order, instead of exponential order. Based on the proposed a posteriori error estimator, we construct an adaptive algorithm for computing the Allen--Cahn equation and its sharp interface limit, the mean curvature flow. Numerical experiments are also presented to show the robustness and effectiveness of the proposed error estimator and the adaptive algorithm.