Cahn-Hilliard方程和Hele-Shaw流的有限元逼近的后验误差估计
A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow
AI总结：
针对Cahn-Hilliard方程和Hele-Shaw流，发展了残差型后验误差估计，并构建了自适应算法，误差界仅以低多项式阶依赖于ε^{-1}。
摘要：
本文针对四阶Cahn-Hilliard方程$u_t+\Deigl(\eps \De u-\eps^{-1} f(u)igr)=0$的协调和混合有限元逼近，发展了残差型后验误差估计。结果表明，后验误差界仅以低多项式阶（而非指数阶）依赖于$\eps^{-1}$。利用这些后验误差估计，我们构建了一个自适应算法来计算Cahn-Hilliard方程及其尖锐界面极限Hele-Shaw流的解。数值实验展示了新误差估计子和所提出的自适应算法的鲁棒性和有效性。
英文摘要：
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.
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