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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
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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Additional Information

Takahito Kashiwabara
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, 153-8914 Tokyo, Japan

Tomoya Kemmochi
Affiliation: Department of Applied Physics, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Aichi, Japan

Keywords: Finite element method, parabolic problems, domain perturbation, analytic semigroup, maximal regularity
Received by editor(s): July 3, 2018
Received by editor(s) in revised form: February 28, 2019, August 16, 2019, and October 7, 2019
Published electronically: January 13, 2020
Additional Notes: The first author was supported by JSPS Kakenhi Grant Number 17K14230, Japan.
The second author was supported by JSPS Kakenhi Grant Number 19K14590, Japan.
Article copyright: © Copyright 2020 American Mathematical Society