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A posteriori error estimates for nonconforming discretizations of singularly perturbed biharmonic operators
Dietmar Gallistl1; Shudan Tian2
1 Friedrich-Schiller-Universität Jena, Institut für Mathematik, Ernst-Abbe-Platz 2, 07743 Jena, Germany
2 School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 355-372.

For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local best-approximation error of the finite element function by piecewise polynomial functions of the degree determining the expected approximation order, which need not coincide with the maximal polynomial degree of the element, for example if bubble functions are used. The error estimator is shown to be reliable and locally efficient up to this polynomial best-approximation error and oscillations of the right-hand side.

Published online:
DOI: 10.5802/smai-jcm.115
Classification: 65N15, 65N30
Keywords: nonconforming finite element, singular perturbation, biharmonic, a posteriori error estimation

Dietmar Gallistl 1; Shudan Tian 2

1 Friedrich-Schiller-Universität Jena, Institut für Mathematik, Ernst-Abbe-Platz 2, 07743 Jena, Germany
2 School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Dietmar Gallistl; Shudan Tian. A posteriori error estimates for nonconforming discretizations of singularly perturbed biharmonic operators. The SMAI Journal of computational mathematics, Volume 10 (2024), pp. 355-372. doi : 10.5802/smai-jcm.115. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.115/

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