Weak discrete maximum principle of isoparametric finite element methods in curvilinear polyhedra
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- by Buyang Li, Weifeng Qiu, Yupei Xie and Wenshan Yu
- Math. Comp. 93 (2024), 1-34
- DOI: https://doi.org/10.1090/mcom/3876
- Published electronically: August 2, 2023
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Abstract:
The weak maximum principle of the isoparametric finite element method is proved for the Poisson equation under the Dirichlet boundary condition in a (possibly concave) curvilinear polyhedral domain with edge openings smaller than $\pi$, which include smooth domains and smooth deformations of convex polyhedra. The proof relies on the analysis of a dual elliptic problem with a discontinuous coefficient matrix arising from the isoparametric finite elements. Therefore, the standard $H^2$ elliptic regularity which is required in the proof of the weak maximum principle in the literature does not hold for this dual problem. To overcome this difficulty, we have decomposed the solution into a smooth part and a nonsmooth part, and estimated the two parts by $H^2$ and $W^{1,p}$ estimates, respectively.
As an application of the weak maximum principle, we have proved a maximum-norm best approximation property of the isoparametric finite element method for the Poisson equation in a curvilinear polyhedron. The proof contains non-trivial modifications of Schatz’s argument due to the nonconformity of the iso-parametric finite elements, which requires us to construct a globally smooth flow map which maps the curvilinear polyhedron to a perturbed larger domain on which we can establish the $W^{1,\infty }$ regularity estimate of the Poisson equation uniformly with respect to the perturbation.
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Bibliographic Information
- 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
- Weifeng Qiu
- Affiliation: Department of Mathematics, City University of Hong Kong, Hung Hom, Hong Kong
- MR Author ID: 845089
- Email: weifeqiu@cityu.edu.hk
- Yupei Xie
- Affiliation: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
- ORCID: 0000-0002-6597-8734
- Email: yupei.xie@polyu.edu.hk
- Wenshan Yu
- Affiliation: Division of Science and Technology, United International College (BNU-HKBU), Zhuhai 519087, People’s Republic of China
- MR Author ID: 1312720
- Email: yuwenshan@uic.edu.cn
- Received by editor(s): February 19, 2022
- Received by editor(s) in revised form: February 22, 2022, and January 11, 2023
- Published electronically: August 2, 2023
- Additional Notes: This work was partially supported by the Research Grants Council of the Hong Kong Special Administrative Region, China (GRF Project No. PolyU15300519, CityU11302219, CityU11300621), and an internal grant at The Hong Kong Polytechnic University (Project ID: P0038843, Work Programme: ZVX7).
The third author is the corresponding author. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 1-34
- MSC (2020): Primary 65N12, 65N30
- DOI: https://doi.org/10.1090/mcom/3876
- MathSciNet review: 4654615