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基于超收敛集群恢复的Cahn-Hilliard方程自适应有限元法
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Abstract:
Cahn-Hilliard方程为四阶非线性的偏微分方程，在物理，生物，化学等各个领域都有广泛的应用，因此研究其数值方法具有实际的应用价值。本文通过分析Cahn-Hilliard方程的一种二阶数值 格式，证明了其误差估计和无条件能量稳定性，并且提出了一个基于后验误差估计的空间和时间 自适应策略，即超收敛集群恢复（superconvergent cluster recovery，简称为SCR）方法，用于数值求解Cahn-Hilliard方程，该策略的主要思想是基于误差估计的结果来控制网格大小，从而可以有效的降低计算成本，最后通过算例证明了SCR 算法的高效性和稳定性。
The Cahn-Hilliard equation is a fourth-order nonlinear partial differential equation with a wide range of applications in various fields such as physics, biology, and chem- istry, so it is of practical application to study its numerical methods. In this study, we analyzed the Cahn-Hilliard equation in a second-order numerical format, demon- strated its error estimate and unconditional energy stability, and suggested a spatial and temporal adaptive strategy based on the posterior error estimate, namely the superconvergent cluster recovery (SCR) method, for numerical solutions.
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