A Posteriori Error Estimates for Finite Element Approximations of the Cahn-hilliard Equation and the Hele-shaw Flow

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut + ∆ ` ε∆u− ε−1f(u) ́ = 0. It is shown that the a posteriori error bounds depends on ε−1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct an adaptive algorithm for computing the solution of the Cahn-Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.