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Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra


Author: Buyang Li
Journal: Math. Comp. 88 (2019), 1-44
MSC (2010): Primary 35K20, 65M12, 65M60
DOI: https://doi.org/10.1090/mcom/3316
Published electronically: March 19, 2018
Previous version: Original version posted March 19, 2018
Corrected version: Current version corrects publisher's error which introduced typos into equations at the bottom of page 17 and top of page 18.
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Abstract: In general polygons and polyhedra, possibly nonconvex, the analyticity of the finite element heat semigroup in the $ L^q$-norm, $ 1\leq q\leq \infty $, and the maximal $ L^p$-regularity of semi-discrete finite element solutions of parabolic equations are proved. By using these results, the problem of maximum-norm stability of the finite element parabolic projection is reduced to the maximum-norm stability of the Ritz projection, which currently is known to hold for general polygonal domains and convex polyhedral domains.


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Additional Information

Buyang Li
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
Email: buyang.li@polyu.edu.hk

DOI: https://doi.org/10.1090/mcom/3316
Keywords: Analytic semigroup, maximal $L^p$-regularity, maximum-norm stability, finite element method, parabolic equation, nonconvex polyhedra
Received by editor(s): December 11, 2017
Received by editor(s) in revised form: May 11, 2017, and July 25, 2017
Published electronically: March 19, 2018
Additional Notes: This work was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region (Project No. 15300817) and by a grant from the Germany/Hong Kong Joint Research Scheme sponsored by the Research Grants Council of Hong Kong and the German Academic Exchange Service of Germany (Ref. No. G-PolyU502/16).
Article copyright: © Copyright 2018 American Mathematical Society

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