Mixed discontinuous Galerkin approximation of the Maxwell operator : the indefinite case
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 4, p. 727-753

We present and analyze an interior penalty method for the numerical discretization of the indefinite time-harmonic Maxwell equations in mixed form. The method is based on the mixed discretization of the curl-curl operator developed in [Houston et al., J. Sci. Comp. 22 (2005) 325-356] and can be understood as a non-stabilized variant of the approach proposed in [Perugia et al., Comput. Methods Appl. Mech. Engrg. 191 (2002) 4675-4697]. We show the well-posedness of this approach and derive optimal a priori error estimates in the energy-norm as well as the ${L}^{2}$-norm. The theoretical results are confirmed in a series of numerical experiments.

DOI : https://doi.org/10.1051/m2an:2005032
Classification:  65N30
Keywords: discontinuous Galerkin methods, mixed methods, time-harmonic Maxwell's equations
@article{M2AN_2005__39_4_727_0,
author = {Houston, Paul and Perugia, Ilaria and Schneebeli, Anna and Sch\"otzau, Dominik},
title = {Mixed discontinuous Galerkin approximation of the Maxwell operator : the indefinite case},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {39},
number = {4},
year = {2005},
pages = {727-753},
doi = {10.1051/m2an:2005032},
zbl = {1087.65106},
mrnumber = {2165677},
language = {en},
url = {http://www.numdam.org/item/M2AN_2005__39_4_727_0}
}

Houston, Paul; Perugia, Ilaria; Schneebeli, Anna; Schötzau, Dominik. Mixed discontinuous Galerkin approximation of the Maxwell operator : the indefinite case. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 4, pp. 727-753. doi : 10.1051/m2an:2005032. http://www.numdam.org/item/M2AN_2005__39_4_727_0/

[1] M. Ainsworth and J. Coyle, Hierarchic $hp$-edge element families for Maxwell’s equations on hybrid quadrilateral/triangular meshes. Comput. Methods Appl. Mech. Engrg. 190 (2001) 6709-6733. | Zbl 0991.78031

[2] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains. Math. Models Appl. Sci. 21 (1998) 823-864. | Zbl 0914.35094

[3] D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749-1779. | Zbl 1008.65080

[4] D. Boffi and L. Gastaldi, Edge finite elements for the approximation of Maxwell resolvent operator. ESAIM: M2AN 36 (2002) 293-305. | Numdam | Zbl 1042.65087

[5] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15, Springer-Verlag, New York (1994). | MR 1278258 | Zbl 0804.65101

[6] Z. Chen, Q. Du and J. Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37 (2000) 1542-1570. | Zbl 0964.78017

[7] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | Zbl 0383.65058

[8] L. Demkowicz and L. Vardapetyan, Modeling of electromagnetic absorption/scattering problems using $hp$-adaptive finite elements. Comput. Methods Appl. Mech. Engrg. 152 (1998) 103-124. | Zbl 0994.78011

[9] P. Fernandes and G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7 (1997) 957-991. | Zbl 0910.35123

[10] R. Hiptmair, Finite elements in computational electromagnetism. Acta Numerica 11 (2002) 237-339. | Zbl 1123.78320

[11] P. Houston, I. Perugia and D. Schötzau, $hp$-DGFEM for Maxwell’s equations, in Numerical Mathematics and Advanced Applications ENUMATH 2001, F. Brezzi, A. Buffa, S. Corsaro, and A. Murli, Eds., Springer-Verlag (2003) 785-794. | Zbl 1061.78012

[12] P. Houston, I. Perugia and D. Schötzau, Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal. 42 (2004) 434-459. | Zbl 1084.65115

[13] P. Houston, I. Perugia and D. Schötzau, Mixed discontinuous Galerkin approximation of the Maxwell operator: Non-stabilized formulation. J. Sci. Comput. 22 (2005) 325-356. | Zbl 1091.78017

[14] P. Houston, I. Perugia, A. Schneebeli and D. Schötzau, Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100 (2005) 485-518. | Zbl 1071.65155

[15] O.A. Karakashian and F. Pascal, A posteriori error estimation for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374-2399. | Zbl 1058.65120

[16] J. L. Lions and E. Magenes, Problèmes aux Limites Non-Homogènes et Applications. Dunod, Paris (1968). | Zbl 0165.10801

[17] P. Monk, A finite element method for approximating the time-harmonic Maxwell equations. Numer. Math. 63 (1992) 243-261. | Zbl 0757.65126

[18] P. Monk, Finite element methods for Maxwell's equations. Oxford University Press, New York (2003). | Zbl 1024.78009

[19] P. Monk, A simple proof of convergence for an edge element discretization of Maxwell's equations, in Computational electromagnetics, C. Carstensen, S. Funken, W. Hackbusch, R. Hoppe and P. Monk, Eds., Springer-Verlag, Lect. Notes Comput. Sci. Engrg. 28 (2003) 127-141. | Zbl 1031.65122

[20] J.C. Nédélec, A new family of mixed finite elements in ${ℝ}^{3}$. Numer. Math. 50 (1986) 57-81. | Zbl 0625.65107

[21] I. Perugia and D. Schötzau, The $hp$-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comput. 72 (2003) 1179-1214. | Zbl 1084.78007

[22] I. Perugia, D. Schötzau and P. Monk, Stabilized interior penalty methods for the time-harmonic Maxwell equations. Comput. Methods Appl. Mech. Engrg. 191 (2002) 4675-4697. | Zbl 1040.78011

[23] A. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comp. 28 (1974) 959-962. | Zbl 0321.65059

[24] L. Vardapetyan and L. Demkowicz, $hp$-adaptive finite elements in electromagnetics. Comput. Methods Appl. Mech. Engrg. 169 (1999) 331-344. | Zbl 0956.78013