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Mathematics of Computation

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Weak discrete maximum principle of finite element methods in convex polyhedra


Authors: Dmitriy Leykekhman and Buyang Li
Journal: Math. Comp. 90 (2021), 1-18
MSC (2010): Primary 65N12, 65N30
DOI: https://doi.org/10.1090/mcom/3560
Published electronically: July 27, 2020
MathSciNet review: 4166450
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Abstract: We prove that the Galerkin finite element solution $ u_h$ of the Laplace equation in a convex polyhedron $ \varOmega $, with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree $ r\geqslant 1$, satisfies the following weak maximum principle:

$\displaystyle \left \Vert u_{h}\right \Vert _{L^{\infty }(\varOmega )} \leqslant C\left \Vert u_{h}\right \Vert _{L^{\infty }(\partial \varOmega )} ,$    

with a constant $ C$ independent of the mesh size $ h$. By using this result, we show that the Ritz projection operator $ R_h$ is stable in $ L^\infty $ norm uniformly in $ h$ for $ r\geq 2$, i.e.,

$\displaystyle \Vert R_hu\Vert _{L^{\infty }(\varOmega )} \leqslant C\Vert u\Vert _{L^{\infty }(\varOmega )} .$    

Thus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains.

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Additional Information

Dmitriy Leykekhman
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
MR Author ID: 680657
Email: dmitriy.leykekhman@uconn.edu

Buyang Li
Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
MR Author ID: 910552
Email: buyang.li@polyu.edu.hk

DOI: https://doi.org/10.1090/mcom/3560
Received by editor(s): September 18, 2019
Received by editor(s) in revised form: February 29, 2020, and April 13, 2020
Published electronically: July 27, 2020
Additional Notes: This work was partially supported by NSF DMS-1913133 and a Hong Kong RGC grant (project no. 15300519).
Article copyright: © Copyright 2020 American Mathematical Society