Cahn-Hilliard方程和Hele-Shaw流的混合内罚不连续Galerkin方法分析
Analysis of mixed interior penalty discontinuous Galerkin methods for the Cahn-Hilliard equation and the Hele-Shaw flow
AI总结：
本文提出并分析了两种完全离散的混合内罚不连续Galerkin方法，用于求解四阶非线性Cahn-Hilliard方程，通过证明数值界面收敛于Hele-Shaw流界面，建立仅依赖于ε^{-1}的误差估计。
摘要：
本文提出并分析了两种完全离散的混合内罚不连续Galerkin方法，用于求解四阶非线性Cahn-Hilliard方程。两种方法均采用后向欧拉方法进行时间离散，内罚不连续Galerkin方法进行空间离散。两者在非线性项的处理方式上有所不同，一种基于完全隐式时间步进，另一种使用能量分裂时间步进。本文的主要目标是证明DG方法的数值界面收敛于Hele-Shaw流界面，通过建立仅依赖于ε^{-1}的误差估计（而非指数级）来实现。类似[14]，关键在于证明离散谱估计在不连续Galerkin有限元空间中的有效性。然而，由于DG空间不是H^1空间的子空间且更大，该结果的有效性不明显。通过精细的扰动论证，利用[14]中证明的有限元空间中的离散谱估计来克服这一困难。此外，还通过数值实验验证理论结果和所提完全离散混合DG方法的性能。
英文摘要：
This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty discontinuous Galerkin methods for spatial discretization. They differ from each other on how the nonlinear term is treated, one of them is based on fully implicit time-stepping and the other uses the energy-splitting time-stepping. The primary goal of the paper is to prove the convergence of the numerical interfaces of the DG methods to the interface of the Hele-Shaw flow. This is achieved by establishing error estimates that depend on $ε^{-1}$ only in some low polynomial orders, instead of exponential orders. Similar to [14], the crux is to prove a discrete spectrum estimate in the discontinuous Galerkin finite element space. However, the validity of such a result is not obvious because the DG space is not a subspace of the (energy) space $H^1$ and it is larger than the finite element space. This difficult is overcome by a delicate perturbation argument which relies on the discrete spectrum estimate in the finite element space proved in \cite{Feng_Prohl04}. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete mixed DG methods.
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