Discrete maximal regularity and the finite element method for parabolic equations
Discrete maximal regularity and the finite element method for parabolic equations
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DOI：
10.1007/s00211-017-0929-z
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发表时间：
2016-02
影响因子：
2.1
通讯作者：
T. Kemmochi;Norikazu Saito
中科院分区：
文献类型：
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作者：
T. Kemmochi;Norikazu Saito
最大正则性是偏微分方程理论中的基本概念。在本文中，我们为多边形或多面体域上的抛物线方程建立了最大正则性的完全离散版本。我们通过质量集总线性热方程导出了有限元近似的离散范数的各种稳定性结果。我们的分析方法是一种使用算子纯虚幂的算子理论分析方法，并且可能是 Dore 和 Venni 结果的离散版本。作为一个应用，证明了这些规范中的最优阶误差估计。此外，我们研究了具有局部 Lipschitz 连续非线性项的半线性热方程的有限元近似，并提供了一种推导最优阶误差估计的新方法。还提出了一些有趣的辅助结果，包括离散 Gagliardo-Nirenberg 和 Sobolev 不等式。
Maximal regularity is a fundamental concept in the theory of partial differential equations. In this paper, we establish a fully discrete version of maximal regularity for parabolic equations on a polygonal or polyhedral domain. We derive various stability results in the discretenorms for the finite element approximation with the mass-lumping to the linear heat equation. Our method of analysis is an operator theoretical one using pure imaginary powers of operators and might be a discrete version of the result of Dore and Venni. As an application, optimal order error estimates in those norms are proved. Furthermore, we study the finite element approximation for semilinear heat equations with locally Lipschitz continuous nonlinear terms and offer a new method for deriving optimal order error estimates. Some interesting auxiliary results including discrete Gagliardo–Nirenberg and Sobolev inequalities are also presented.