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A Potential-Robust WG Finite Element Method for the Maxwell Equations on Tetrahedral Meshes

  • Yufeng Nie , Xiu Ye and Shangyou Zhang ORCID logo EMAIL logo
Published/Copyright: February 14, 2025

Abstract

A potential-robust WG (weak Galerkin) method is introduced for the Maxwell equations. We obtain potential/magnetic-permeability independent error estimates. Optimal-order convergence rates are proved in both the energy norm and the L 2 norm. Numerical examples verify the theory.

MSC 2020: 65N15; 65N30

Award Identifier / Grant number: 11971386

Award Identifier / Grant number: 2020YFA0713603

Funding statement: Yufeng Nie was supported in part by National Natural Science Foundation of China (No. 11971386) and the National Key R&D Program of China (No. 2020YFA0713603).

References

[1] A. Bossavit, Computational Electromagnetism, Academic Press, San Diego, 1998. Search in Google Scholar

[2] S. C. Brenner, J. Cui, Z. Nan and L.-Y. Sung, Hodge decomposition for divergence-free vector fields and two-dimensional Maxwell’s equations, Math. Comp. 81 (2012), no. 278, 643–659. 10.1090/S0025-5718-2011-02540-8Search in Google Scholar

[3] S. C. Brenner, F. Li and L.-Y. Sung, A locally divergence-free interior penalty method for two-dimensional curl-curl problems, SIAM J. Numer. Anal. 46 (2008), no. 3, 1190–1211. 10.1137/060671760Search in Google Scholar

[4] S. Caorsi, P. Fernandes and M. Raffetto, On the convergence of Galerkin finite element approximations of electromagnetic eigenproblems, SIAM J. Numer. Anal. 38 (2000), no. 2, 580–607. 10.1137/S0036142999357506Search in Google Scholar

[5] H. Duan, Z. Du, W. Liu and S. Zhang, New mixed elements for Maxwell equations, SIAM J. Numer. Anal. 57 (2019), no. 1, 320–354. 10.1137/18M1168054Search in Google Scholar

[6] H. Duan, W. Liu, J. Ma, R. C. E. Tan and S. Zhang, A family of optimal Lagrange elements for Maxwell’s equations, J. Comput. Appl. Math. 358 (2019), 241–265. 10.1016/j.cam.2019.03.022Search in Google Scholar

[7] P. Houston, I. Perugia and D. Schötzau, hp-DGFEM for Maxwell’s equations, Numerical Mathematics and Advanced Applications, Springer, Milan (2003), 785–794. 10.1007/978-88-470-2089-4_71Search in Google Scholar

[8] P. Houston, I. Perugia and D. Schötzau, Mixed discontinuous Galerkin approximation of the Maxwell operator, Technical Report 02-16, University of Basel, Basel, 2002. Search in Google Scholar

[9] P. Houston, I. Perugia and D. Schötzau, Mixed discontinuous Galerkin approximation of the Maxwell operator, SIAM J. Numer. Anal. 42 (2004), no. 1, 434–459. 10.1137/S003614290241790XSearch in Google Scholar

[10] J. Huang and S. Zhang, A divergence-free finite element method for a type of 3D Maxwell equations, Appl. Numer. Math. 62 (2012), no. 6, 802–813. 10.1016/j.apnum.2011.06.009Search in Google Scholar

[11] J. Jin, The Finite Element Method in Electromagnetics, 2nd ed., Wiley-Interscience, New York, 2002. Search in Google Scholar

[12] J. Li, C. Shi and C.-W. Shu, Optimal non-dissipative discontinuous Galerkin methods for Maxwell’s equations in Drude metamaterials, Comput. Math. Appl. 73 (2017), no. 8, 1760–1780. 10.1016/j.camwa.2017.02.018Search in Google Scholar

[13] P. Monk, Finite Element Methods for Maxwell’s Equations, Numer. Math. Sci. Comput., Oxford University, New York, 2003. 10.1093/acprof:oso/9780198508885.001.0001Search in Google Scholar

[14] P. Monk and S. Zhang, Multigrid computation of vector potentials, J. Comput. Appl. Math. 62 (1995), no. 3, 301–320. 10.1016/0377-0427(94)00106-8Search in Google Scholar

[15] L. Mu, J. Wang, X. Ye and S. Zhang, A weak Galerkin finite element method for the Maxwell equations, J. Sci. Comput. 65 (2015), no. 1, 363–386. 10.1007/s10915-014-9964-4Search in Google Scholar

[16] J.-C. Nédélec, Mixed finite elements in 𝐑 3 , Numer. Math. 35 (1980), no. 3, 315–341. 10.1007/BF01396415Search in Google Scholar

[17] J.-C. Nédélec, A new family of mixed finite elements in 𝐑 3 , Numer. Math. 50 (1986), no. 1, 57–81. 10.1007/BF01389668Search in Google Scholar

[18] I. Perugia and D. Schötzau, The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations, Math. Comp. 72 (2003), no. 243, 1179–1214. 10.1090/S0025-5718-02-01471-0Search in Google Scholar

[19] I. Perugia, D. Schötzau and P. Monk, Stabilized interior penalty methods for the time-harmonic Maxwell equations, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 41–42, 4675–4697. 10.1016/S0045-7825(02)00399-7Search in Google Scholar

[20] S. Shields, J. Li and E. A. Machorro, Weak Galerkin methods for time-dependent Maxwell’s equations, Comput. Math. Appl. 74 (2017), no. 9, 2106–2124. 10.1016/j.camwa.2017.07.047Search in Google Scholar

[21] L. Vardapetyan and L. Demkowicz, hp-adaptive finite elements in electromagnetics, Comput. Methods Appl. Mech. Engrg. 169 (1999), no. 3–4, 331–344. 10.1016/S0045-7825(98)00161-3Search in Google Scholar

[22] C. Wang, X. Ye and S. Zhang, A modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes, J. Comput. Appl. Math. 448 (2024), Article ID 115918. 10.1016/j.cam.2024.115918Search in Google Scholar

[23] J. Wang and X. Ye, New finite element methods in computational fluid dynamics by H ( div ) elements, SIAM J. Numer. Anal. 45 (2007), no. 3, 1269–1286. 10.1137/060649227Search in Google Scholar

[24] J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math. 241 (2013), 103–115. 10.1016/j.cam.2012.10.003Search in Google Scholar

[25] X. Ye and S. Zhang, A stabilizer-free weak Galerkin finite element method on polytopal meshes, J. Comput. Appl. Math. 371 (2020), Article ID 112699. 10.1016/j.cam.2019.112699Search in Google Scholar

[26] X. Ye and S. Zhang, A stabilizer free weak Galerkin method for the biharmonic equation on polytopal meshes, SIAM J. Numer. Anal. 58 (2020), no. 5, 2572–2588. 10.1137/19M1276601Search in Google Scholar

[27] X. Ye and S. Zhang, A new weak gradient for the stabilizer free weak Galerkin method with polynomial reduction, Discrete Contin. Dyn. Syst. Ser. B 26 (2021), no. 8, 4131–4145. 10.3934/dcdsb.2020277Search in Google Scholar

[28] X. Ye and S. Zhang, A numerical scheme with divergence free H - div triangular finite element for the Stokes equations, Appl. Numer. Math. 167 (2021), 211–217. 10.1016/j.apnum.2021.05.005Search in Google Scholar

[29] X. Ye and S. Zhang, A stabilizer-free pressure-robust finite element method for the Stokes equations, Adv. Comput. Math. 47 (2021), no. 2, Paper No. 28. 10.1007/s10444-021-09856-9Search in Google Scholar

[30] X. Ye and S. Zhang, A stabilizer free weak Galerkin finite element method on polytopal mesh: Part II, J. Comput. Appl. Math. 394 (2021), Article ID 113525. 10.1016/j.cam.2021.113525Search in Google Scholar

[31] X. Ye and S. Zhang, A stabilizer free weak Galerkin finite element method on polytopal mesh: Part III, J. Comput. Appl. Math. 394 (2021), Article ID 113538. 10.1016/j.cam.2021.113538Search in Google Scholar

[32] X. Ye and S. Zhang, A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh, Electron. Res. Arch. 29 (2021), no. 6, 3609–3627. 10.3934/era.2021053Search in Google Scholar

[33] X. Ye and S. Zhang, Achieving superconvergence by one-dimensional discontinuous finite elements: Weak Galerkin method, East Asian J. Appl. Math. 12 (2022), no. 3, 590–598. 10.4208/eajam.030921.141121Search in Google Scholar

[34] X. Ye and S. Zhang, Constructing order two superconvergent WG finite elements on rectangular meshes, Numer. Math. Theory Methods Appl. 16 (2023), no. 1, 230–241. 10.4208/nmtma.OA-2022-0082Search in Google Scholar

[35] S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp. 74 (2005), no. 250, 543–554. 10.1090/S0025-5718-04-01711-9Search in Google Scholar

[36] S. Zhang, On the P1 Powell–Sabin divergence-free finite element for the Stokes equations, J. Comput. Math. 26 (2008), no. 3, 456–470. Search in Google Scholar

[37] S. Zhang, A family of Q k + 1 , k × Q k , k + 1 divergence-free finite elements on rectangular grids, SIAM J. Numer. Anal. 47 (2009), no. 3, 2090–2107. 10.1137/080728949Search in Google Scholar

[38] S. Zhang, Divergence-free finite elements on tetrahedral grids for k 6 , Math. Comp. 80 (2011), no. 274, 669–695. 10.1090/S0025-5718-2010-02412-3Search in Google Scholar

[39] S. Zhang, Quadratic divergence-free finite elements on Powell–Sabin tetrahedral grids, Calcolo 48 (2011), no. 3, 211–244. 10.1007/s10092-010-0035-4Search in Google Scholar

Received: 2024-09-10
Revised: 2024-12-08
Accepted: 2025-02-09
Published Online: 2025-02-14

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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