A Least-Squares-Based Weak Galerkin Finite-Element Method for the Time-Harmonic Maxwell’s Equations

Abstract

In this article, we propose and analyze a least-squares-based weak Galerkin finite-element method (WG-FEM) for solving the indefinite time-harmonic Maxwell’s equations in

{\mathbb{R}}^d(d=2,3).

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^d\;(d=2, 3).$$\end{document}

The super-convergence of order one for the discrete

{\mathbf{H}}^1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textbf{H}^1$$\end{document}

-like norm has been established. Numerical simulations show that the approximate solutions converge to the exact solutions with optimal rates in the

L^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}

norm on hybrid meshes. In addition, this method is shown to be absolutely stable under low regularity requirements with high wave numbers.

Keywords

Least-squares / Weak Galerkin (WG) / Finite-element method / Maxwell’s equations / Polygonal/Polyhedral meshes / Primary: 65N15 / 65N30 / 76D07 / Secondary: 35B45 / 35J50

References

[1] Adams, R.A., Fournier, J.: Sobolev Spaces. Academic Press, New York (1975)

[2] Ainsworth, M., Coyle, J.: Hierarchic hp-edge element families for Maxwell’s equations on hybrid quadrilateral/triangular meshes. Comput. Methods Appl. Mech. Eng. 190(49/50), 6709–6733 (2001)

[3] Alvarado, A., Castillo, P.: Computational performance of LDG methods applied to time-harmonic Maxwell’s equation in polyhedral domains. J. Sci. Comput. 67, 453–474 (2016)

[4] Bacuta, C., Jacavage, J., Qirko, K., Sayas, F. J.: Saddle point least squares iterative solvers for the time harmonic Maxwell’s equations. Comput. Math. Appl. 74(11), 2915–2928 (2017)

[5] Bensow, R., Larson, M.G.: Discontinuous least-squares finite element method for the div-curl problem. Numer. Math. 101(4), 601–617 (2005)

[6] Bochev, P., Lai, J., Olson, L.: A locally conservative, discontinuous least-squares finite element method for the Stokes equations. International Journal for Numerical Methods in Fluids 68(6), 782–804 (2012)

[7] Bochev, P.B., Gunzburger, M.D.: Finite element methods of least-squares type. SIAM Review 40(4), 789–837 (1998)

[8] Bramble, J.H., Kolev, T.V., Pasciak, J.E.: A least-squares approximation method for the time-harmonic Maxwell equations. Journal of Numerical Mathematics 13(4) 237–263 (2005)

[9] Brenner, S, Li, F, Sung, L Y.: A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell’s equations. Math. Comput. 76(258), 573–595 (2007)

[10] Burman, E., Duran, O., Ern, A., Steins, M.: Convergence analysis of hybrid high-order methods for the wave equation. J. Sci. Comput. 87(3), 1–30 (2021)

[11] Carstensen, C., Zhai, Q., Zhang, R.: A skeletal finite element method can compute lower eigenvalue bounds. SIAM J. Numer. Anal. 58(1), 109–124 (2020)

[12] Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2000)

[13] Chang CL. A least-squares finite element method for the Helmholtz equation. Comput. Methods Appl. Mech. Eng., 1990, 83(1): 1-7

[14] Chen, H., Qiu, W.: A first order system least squares method for the Helmholtz equation. Journal of Computational and Applied Mathematics 309, 145–162 (2017)

[15] Cicuttin M., Geuzaine C.: Numerical investigation of a 3D hybrid high-order method for the indefinite time-harmonic Maxwell’s problem. Finite Element in Analysis and Design 233, 104124 (2024)

[16] Cockburn, B., Di Pietro, D.A., Ern, A.: Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: Mathematical Modeling and Numerical Analysis 50(3), 635–650 (2016)

[17] Dong, Z., Ern, A.: Hybrid high-order and weak Galerkin methods for the biharmonic problem. arXiv:2103.16404 (2021)

[18] Egger H, Schöberl J. A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems. IMA J. Numer. Anal., 2010, 30(4): 1206-1234

[19] Feng, X., Lu, P., Xu, X. A hybridizable discontinuous Galerkin method for the time-harmonic Maxwell’s equations with high wave number. Computational Methods in Applied Mathematics 16(3), 429–445 (2016)

[20] Feng, X., Wu, H.J.: An absolutely stable discontinuous Galerkin method for the indefinite time-harmonic Maxwell’s equations with large wave number. SIAM J. Numer. Anal. 52(5), 2356–2380 (2014)

[21] Vivette, G., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, volume 5. Springer Science and Business Media (2012)

[22] Graglia, R.D.: Donald R Wilton, D.R., Peterson, A.F.: Higher order interpolatory vector bases for computational electromagnetics. IEEE Transactions on Antennas and Propagation 45(3), 329–342 (1997)

[23] Gudi, T., Mallik, G., Pramanick, T.: A hybrid high-order method for quasilinear elliptic problems of nonmonotone type. SIAM J. Numer. Anal. 60(4), 2318–2344 (2022)

[24] Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numerica 11, 237–339 (2002)

[25] Hiptmair, R., Moiola, A., Perugia, I.: Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell’s equations. Math. Comput. 82(281), 247–268 (2013)

[26] Houston, Perugia, P., I., Schneebeli, A., Schötzau, D.: Interior penalty method for the indefinite time-harmonic Maxwell’s equations. Numer. Math. 100, 485–518 (2005)

[27] Houston, P., Perugia, I., Schneebeli, A., Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell’s operator: the indefinite case. ESAIM: Mathematical Modelling and Numerical Analysis, 39(4),727–753 (2005)

[28] Hu, Q., Song, R.: A variant of the plane wave least squares method for the time-harmonic Maxwell’s equations. ESAIM: Mathematical Modelling and Numerical Analysis 53(1), 85–103 (2019)

[29] Hughes, T.J.R., Masud, A., Wan, J.:. A stabilized mixed discontinuous Galerkin method for Darcy flow. Comput. Methods Appl. Mech. Eng., 195(25/26/27/28), 3347–3381 (2006)

[30] Jin, J.-M.:. The Finite Element Method in Electromagnetics. John Wiley & Sons (2015)

[31] Kumar, R., Deka, B.: Weak Galerkin finite element methods for H(curl) and H(curl, div)-elliptic problems. Comput. Math. Appl. 147, 210–221 (2023)

[32] Li J, Ye X, Zhang S. A weak Galerkin least-squares finite element method for div-curl systems. J. Comput. Phys., 2018, 363: 79-86

[33] Li R, Liu Q, Yang F. A discontinuous least squares finite element method for the Helmholtz equation. Numerical Methods for Partial Differential Equations, 2023, 39(2): 1425-1448

[34] Li R, Yang F. A sequential least squares method for Poisson equation using a patch reconstructed space. SIAM J. Numer. Anal., 2020, 58(1): 353-374

[35] Lu, P., Chen, H., Qiu, W.: An absolutely stable hp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$hp$$\end{document}-HDG method for the time-harmonic Maxwell’s equations with high wave number. Math. Comput. 86(306), 1553–1577 (2017)

[36] Monk P. A finite element method for approximating the time-harmonic Maxwell’s equations. Numer. Math., 1992, 63(1): 243-261

[37] Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)

[38] Lin M, Wang J, Ye X. A least-squares-based weak Galerkin finite element method for second order elliptic equations. SIAM J. Sci. Comput., 2017, 39(4): A1531-A1557

[39] Lin M, Wang J, Ye X, Zhang S. A weak Galerkin finite element method for the Maxwell’s equations. J. Sci. Comput., 2015, 65(1): 363-386

[40] Nédélec, J.C.: A new family of mixed finite elements in R3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb{R} }^3$$\end{document}. Numerische Mathematik 50, 57–81 (1986)

[41] Nguyen, N.C., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations. J. Comput. Phys., 230(19), 7151–7175 (2011)

[42] Schöberl J, Zaglmayr S. High order Nédélec elements with local complete sequence properties. COMPEL, 2005, 24(2): 374-384

[43] Tang, M., Zhong, L., Xie, Y.: A modified weak Galerkin method for H(curl)-elliptic problem. Comput. Math. Appl., 139, 224–229 (1987)

[44] Wang C. New discretization schemes for time-harmonic Maxwell’s equations by weak Galerkin finite element methods. J. Comput. Appl. Math., 2018, 341: 127-143

[45] Wang C, Wang J. Discretization of div-curl systems by weak Galerkin finite element methods on polyhedral partitions. J. Sci. Comput., 2016, 68(3): 1144-1171

[46] Wang J, Ye X. A weak Galerkin mixed finite element method for second order elliptic problems. Math. Comput., 2014, 83(289): 2101-2126

[47] Wang X, Ye X, Zhang S, Zhu P. A weak Galerkin least squares finite element method of Cauchy problem for Poisson equation. J. Comput. Appl. Math., 2022, 401 113767

[48] Xie Y, Tang M, Tang C. A weak Galerkin finite element method for indefinite time-harmonic Maxwell equations. Appl. Math. Comput., 2022, 435127471

[49] Zhu L, Huang T-Z, Li L. A hybrid-mesh hybridizable discontinuous Galerkin method for solving the time-harmonic Maxwell’s equations. Appl. Math. Lett., 2017, 68: 109-116

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Shanghai University