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

2007/08/15 by Xiaobing Feng, Feng, Xiaobing, Haijun Wu +1 · 1 citation
Computer Science · Engineering · Materials Science · #53A10 #65M12 #65M15 #65M60 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.0708.2116

openalex publication_date 2007/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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(\eps \De u-\eps^-1 f(u))=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.

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