A posteriori error estimates and an adaptive finite element method for the Allen-Cahn equation and the mean curvature flow

This paper develops an a posteriori error estimate of residual type for finite element approximations of the Allen - Cahn equation u(t) - Delta u + epsilon(-2)f ( u) = 0. It is shown that the error depends on epsilon(-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.