Volume 26, Issue 6
A Posteriori Error Estimates for Finite Element Approximations of the Cahn-Hilliard Equation and the Hele-Shaw Flow
DOI:

J. Comp. Math., 26 (2008), pp. 767-796

Published online: 2008-12

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• 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 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.

• Keywords

Cahn-Hilliard equation Hele-Shaw flow Phase transition Conforming elements Mixed finite element methods A posteriori error estimates Adaptivity

65M60 65M12 65M15 53A10

@Article{JCM-26-767, author = {}, title = {A Posteriori Error Estimates for Finite Element Approximations of the Cahn-Hilliard Equation and the Hele-Shaw Flow}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {6}, pages = {767--796}, 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 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.}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8659.html} }
TY - JOUR T1 - A Posteriori Error Estimates for Finite Element Approximations of the Cahn-Hilliard Equation and the Hele-Shaw Flow JO - Journal of Computational Mathematics VL - 6 SP - 767 EP - 796 PY - 2008 DA - 2008/12 SN - 26 DO - http://dor.org/ UR - https://global-sci.org/intro/jcm/8659.html KW - Cahn-Hilliard equation KW - Hele-Shaw flow KW - Phase transition KW - Conforming elements KW - Mixed finite element methods KW - A posteriori error estimates KW - Adaptivity AB - 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 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.