Comptes Rendus
Numerical analysis
Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence
Comptes Rendus. Mathématique, Volume 361 (2023), pp. 723-736.

We estimate best-approximation errors using vector-valued finite elements for fields with low regularity in the scale of the fractional-order Sobolev spaces. By assuming that the target field enjoys an additional integrability property on its curl or its divergence, we establish upper bounds on these errors that can be localized to the mesh cells. These bounds are derived using the quasi-interpolation errors with or without boundary prescription derived in [A. Ern and J.-L. Guermond, ESAIM Math. Model. Numer. Anal., 51 (2017), pp. 1367–1385]. In the present work, a localized upper bound on the quasi-interpolation error is derived by using the face-to-cell lifting operators analyzed in [A. Ern and J.-L. Guermond, Found. Comput. Math., (2021)] and by exploiting the additional assumption made on the curl or the divergence of the target field. As an illustration, we show how to apply these results to the error analysis of the curl-curl problem associated with Maxwell’s equations.

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Published online:
DOI: 10.5802/crmath.347
Classification: 65D05, 65N30, 41A65

Zhaonan Dong 1, 2; Alexandre Ern 2, 1; Jean-Luc Guermond 3

1 Inria, 2 rue Simone Iff, 75589 Paris, France
2 CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée, France
3 Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence},
     journal = {Comptes Rendus. Math\'ematique},
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Zhaonan Dong; Alexandre Ern; Jean-Luc Guermond. Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 723-736. doi : 10.5802/crmath.347. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.347/

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