Geodesic
Analyse numérique
Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence
Dong, Zhaonan1, 2  ; Ern, Alexandre2, 1 ; Guermond, Jean-Luc3
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
Comptes Rendus. Mathématique, Tome 361 (2023) no. G4, pp. 723-736

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

Dong, Zhaonan 1, 2 ; Ern, Alexandre 2, 1 ; Guermond, Jean-Luc 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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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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Dong, Zhaonan; Ern, Alexandre; Guermond, Jean-Luc. 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, Tome 361 (2023) no. G4, pp. 723-736. doi: 10.5802/crmath.347

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