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://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 - <p style="text-align: justify;">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.</p>