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

Xiaobing Feng & Haijun Wu

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 $u_t+∆(ε∆u−ε^{-1}f(u))=0$. 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.

  • Keywords

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

  • AMS Subject Headings

65M60, 65M12, 65M15, 53A10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-26-767, author = {Xiaobing Feng and Haijun Wu}, 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 $u_t+∆(ε∆u−ε^{-1}f(u))=0$. 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.

}, 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 AU - Xiaobing Feng & Haijun Wu JO - Journal of Computational Mathematics VL - 6 SP - 767 EP - 796 PY - 2008 DA - 2008/12 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8659.html KW - Cahn-Hilliard equation, Hele-Shaw flow, Phase transition, Conforming elements, Mixed finite element methods, A posteriori error estimates, 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 $u_t+∆(ε∆u−ε^{-1}f(u))=0$. 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.

Xiaobing Feng and Haijun Wu. (2008). A Posteriori Error Estimates for Finite Element Approximations of the Cahn-Hilliard Equation and the Hele-Shaw Flow. Journal of Computational Mathematics. 26 (6). 767-796. doi:
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