A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow

TL;DR

This paper introduces residual-based a posteriori error estimates for finite element methods solving the Cahn-Hilliard equation, enabling adaptive algorithms that efficiently handle the sharp interface limit with controlled error dependence on parameters.

Contribution

It develops new a posteriori error bounds with polynomial dependence on ps^{-1} and constructs an adaptive algorithm for the Cahn-Hilliard and Hele-Shaw flow equations.

Findings

Error bounds depend polynomially on ps^{-1}

Adaptive algorithm effectively computes solutions with controlled error

Numerical experiments confirm robustness and efficiency

Abstract

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation $u_t+\De\bigl(\eps \De u-\eps^{-1} f(u)\bigr)=0$. It is shown that the {\it a posteriori} error bounds depends on $\eps^{-1}$ 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.