Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence
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 thes...
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Académie des sciences
2023-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.347/ |
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| author | Dong, Zhaonan Ern, Alexandre Guermond, Jean-Luc |
| author_facet | Dong, Zhaonan Ern, Alexandre Guermond, Jean-Luc |
| author_sort | Dong, Zhaonan |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-24458807064a4dff8273546773d53cdf |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-24458807064a4dff8273546773d53cdf2025-02-07T11:07:37ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-05-01361G472373610.5802/crmath.34710.5802/crmath.347Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergenceDong, Zhaonan0Ern, Alexandre1Guermond, Jean-Luc2CERMICS, Ecole des Ponts, 77455 Marne-la-Vallée, France; Inria, 2 rue Simone Iff, 75589 Paris, FranceCERMICS, Ecole des Ponts, 77455 Marne-la-Vallée, France; Inria, 2 rue Simone Iff, 75589 Paris, FranceDepartment of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843, USAWe 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.347/ |
| spellingShingle | 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 |
| title | Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence |
| title_full | Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence |
| title_fullStr | Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence |
| title_full_unstemmed | Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence |
| title_short | Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence |
| title_sort | local decay rates of best approximation errors using vector valued finite elements for fields with low regularity and integrable curl or divergence |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.347/ |
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