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 $u_t+\text{De}(\text{eps}\text{De}u-{\text{eps}}^{-1}f(u))=0$. It is shown that the {\it a posteriori} error bounds depends on ${\text{eps}}^{-1}$ 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.