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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@article{CRMATH_2023__361_G4_723_0,
author = {Zhaonan Dong and Alexandre Ern and Jean-Luc Guermond},
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},
pages = {723--736},
publisher = {Acad\'emie des sciences, Paris},
volume = {361},
year = {2023},
doi = {10.5802/crmath.347},
language = {en},
}
TY - JOUR AU - Zhaonan Dong AU - Alexandre Ern AU - Jean-Luc Guermond TI - Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence JO - Comptes Rendus. Mathématique PY - 2023 SP - 723 EP - 736 VL - 361 PB - Académie des sciences, Paris DO - 10.5802/crmath.347 LA - en ID - CRMATH_2023__361_G4_723_0 ER -
%0 Journal Article %A Zhaonan Dong %A Alexandre Ern %A Jean-Luc Guermond %T Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence %J Comptes Rendus. Mathématique %D 2023 %P 723-736 %V 361 %I Académie des sciences, Paris %R 10.5802/crmath.347 %G en %F CRMATH_2023__361_G4_723_0
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/
[1] An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comput., Volume 68 (1999) no. 226, pp. 607-631 | DOI | MR | Zbl
[2] Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., Volume 21 (1998) no. 9, pp. 823-864 | DOI | MR | Zbl
[3] Numerical treatment of realistic boundary conditions for the eddy current problem in an electrode via Lagrange multipliers, Math. Comput., Volume 74 (2005) no. 249, pp. 123-151 | DOI | MR | Zbl
[4] -theory of the Maxwell operator in arbitrary domains, Russ. Math. Surv., Volume 42 (1987) no. 6, p. 75 | DOI | Zbl
[5] Mixed finite element methods and applications, Springer Series in Computational Mathematics, 44, Springer, 2013 | DOI | Zbl
[6] Interpolation estimates for edge finite elements and application to band gap computation, Appl. Numer. Math., Volume 56 (2006) no. 10-11, pp. 1283-1292 | DOI | MR | Zbl
[7] Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains, J. Math. Anal. Appl., Volume 408 (2013) no. 259, pp. 498-512 | DOI | MR | Zbl
[8] Fully discrete finite element approaches for time-dependent Maxwell’s equations, Numer. Math., Volume 82 (1999) no. 2, pp. 193-219 | DOI | MR | Zbl
[9] A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains, Math. Methods Appl. Sci., Volume 12 (1990) no. 4, pp. 365-368 | DOI | MR | Zbl
[10] Finite element quasi-interpolation and best approximation, ESAIM, Math. Model. Numer. Anal., Volume 51 (2017) no. 4, pp. 1367-1385 | MR | Zbl
[11] Abstract nonconforming error estimates and application to boundary penalty methods for diffusion equations and time-harmonic Maxwell’s equations, Comput. Methods Appl. Math., Volume 18 (2018) no. 3, pp. 451-475 | DOI | MR | Zbl
[12] Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions, Comput. Math. Appl., Volume 75 (2018) no. 3, pp. 918-932 | DOI | MR | Zbl
[13] Finite Elements I: Approximation and Interpolation, Texts in Applied Mathematics, 72, Springer, 2020 | Zbl
[14] Finite Elements II. Galerkin Approximation, Elliptic and Mixed PDEs, Texts in Applied Mathematics, 73, Springer, 2021 | DOI | Zbl
[15] Quasi-optimal nonconforming approximation of elliptic PDEs with contrasted coefficients and , , regularity, Found. Comput. Math., Volume 22 (2022) no. 5, pp. 1273-1308 | DOI | Zbl
[16] Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 22, Masson, 1992 | Zbl
[17] Regularity of weak solutions of Maxwell’s equations with mixed boundary-conditions, Math. Methods Appl. Sci., Volume 22 (1999) no. 14, pp. 1255-1274 | DOI | MR | Zbl
[18] Finite element methods for Maxwell’s equations, Numerical Mathematics and Scientific Computation, Oxford University Press, 2003 | DOI | Zbl
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