Error analysis of a mixed finite element method for a Cahn–Hilliard–Hele–Shaw system

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• 作者：Yuan Liu ; Wenbin Chen ; Cheng Wang ; Steven M. Wise
• 关键词：Mathematics Subject Classification35K35 ; 35K55 ; 65K10 ; 65M12 ; 65M60
• 刊名：Numerische Mathematik
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：135
• 期：3
• 页码：679-709
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Mathematics, general; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation; Appl.Mathematics/Comput
• 出版者：Springer Berlin Heidelberg
• ISSN：0945-3245
• 卷排序：135

文摘

We present and analyze a mixed finite element numerical scheme for the Cahn–Hilliard–Hele–Shaw equation, a modified Cahn–Hilliard equation coupled with the Darcy flow law. This numerical scheme was first reported in Feng and Wise (SIAM J Numer Anal 50:1320–1343, 2012), with the weak convergence to a weak solution proven. In this article, we provide an optimal rate error analysis. A convex splitting approach is taken in the temporal discretization, which in turn leads to the unique solvability and unconditional energy stability. Instead of the more standard \(\ell ^\infty (0,T;L^2) \cap \ell ^2 (0,T; H^2)\) error estimate, we perform a discrete \(\ell ^\infty (0,T; H^1) \cap \ell ^2 (0,T; H^3 )\) error estimate for the phase variable, through an \(L^2\) inner product with the numerical error function associated with the chemical potential. As a result, an unconditional convergence (for the time step \(\tau \) in terms of the spatial resolution h) is derived. The nonlinear analysis is accomplished with the help of a discrete Gagliardo–Nirenberg type inequality in the finite element space, gotten by introducing a discrete Laplacian \(\Delta _h\) of the numerical solution, such that \(\Delta _h \phi \in S_h\), for every \(\phi \in S_h\), where \(S_h\) is the finite element space.