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Discrete maximal regularity and the finite element method for parabolic equations

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Abstract

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 \(\varOmega \). We derive various stability results in the discrete \(L^p(0,T;L^q(\varOmega ))\) norms 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.

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Acknowledgements

The first author was supported by the Program for Leading Graduate Schools, MEXT, Japan and JSPS KAKENHI Grant No. 15J07471, Japan. The second author was supported by JST CREST Grant No. JPMJCR15D1, Japan and by JSPS KAKENHI Grant Nos. 15H03635 and 15K13454, Japan. The authors would like to thank the anonymous reviewer for valuable comments and suggestions to improve the quality of the paper.

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  1. Graduate School of Mathematical Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo, 153-8914, Japan

    Tomoya Kemmochi & Norikazu Saito

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  1. Tomoya Kemmochi
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  2. Norikazu Saito
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Correspondence to Tomoya Kemmochi.

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Kemmochi, T., Saito, N. Discrete maximal regularity and the finite element method for parabolic equations. Numer. Math. 138, 905–937 (2018). https://doi.org/10.1007/s00211-017-0929-z

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  • DOI: https://doi.org/10.1007/s00211-017-0929-z

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