Stability, analyticity, and maximal regularity for parabolic finite element problems on smooth domains

Kashiwabara, Takahito; Kemmochi, Tomoya

doi:10.1090/mcom/3500

Stability, analyticity, and maximal regularity for parabolic finite element problems on smooth domains

Authors: Takahito Kashiwabara and Tomoya Kemmochi
Journal: Math. Comp. 89 (2020), 1647-1679
MSC (2010): Primary 65M60; Secondary 65M12, 65M15
DOI: https://doi.org/10.1090/mcom/3500
Published electronically: January 13, 2020
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Abstract: In this paper, we consider the finite element semidiscretization for a parabolic problem on a smooth domain $\Omega \subset \mathbb{R}^N$ with the Neumann boundary condition. We emphasize that the domain can be nonconvex in general. We discretize the this problem by the finite element method by constructing a family of polygonal or polyhedral domains $\{ \Omega _h \}_h$ that approximate the original domain $\Omega$ . The aim of this study is to derive the smoothing property for the discrete parabolic semigroup and the maximal regularity for the discrete elliptic operator. The main difficulty is the effect of the boundary-skin (symmetric difference) $\Omega \bigtriangleup \Omega _h$ . In order to address the effect of the boundary-skin, we introduce the tubular neighborhood of the original boundary $\partial \Omega$ .

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