Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, p. 413-430
In this paper we prove the discrete compactness property for a discontinuous Galerkin approximation of Maxwell's system on quite general tetrahedral meshes. As a consequence, a discrete Friedrichs inequality is obtained and the convergence of the discrete eigenvalues to the continuous ones is deduced using the theory of collectively compact operators. Some numerical experiments confirm the theoretical predictions.
DOI : https://doi.org/10.1051/m2an:2006017
Classification:  65N25,  65N30
Keywords: DG method, Maxwell's system, discrete compactness, eigenvalue approximation
@article{M2AN_2006__40_2_413_0,
     author = {Creus\'e, Emmanuel and Nicaise, Serge},
     title = {Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {2},
     year = {2006},
     pages = {413-430},
     doi = {10.1051/m2an:2006017},
     zbl = {1112.78020},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_2_413_0}
}
Creusé, Emmanuel; Nicaise, Serge. Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 2, pp. 413-430. doi : 10.1051/m2an:2006017. http://www.numdam.org/item/M2AN_2006__40_2_413_0/

[1] P. Anselone, Collectively compact operator approximation theory. Prentice Hall (1971). | MR 443383

[2] T. Apel and S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Method. Appl. Sci. 21 (1998) 519-549. | Zbl 0911.65107

[3] P. Arbenz and R. Geus, Eigenvalue solvers for electromagnetic fields in cavities, in High performance scientific and engineering computing, H.-J. Bungartz, F. Durst, and C. Zenger, Eds., Lect. Notes Comput. Sc., Springer, Berlin 8(1999). | MR 1715227

[4] D.G. 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

[5] F. Assous, P. Ciarlet and E. Sonnendrücker, Characterization of the singular part of the solution of Maxwell's equations in a polyhedral domain. RAIRO Modél. Math. Anal. Numér. 32 (1998) 485-499. | Zbl 0931.35169

[6] M. Birman and M. Solomyak, L 2 -theory of the Maxwell operator in arbitrary domains. Russ. Math. Surv. 42 (1987) 75-96. | Zbl 0653.35075

[7] D. Boffi, Fortin operator and discrete compactness for edge elements. Numer. Math. 87 (2000) 229-246. | Zbl 0967.65106

[8] D. Boffi, L. Demkowicz and M. Costabel, Discrete compactness for p and hp 2d edge finite elements. Math. Mod. Meth. Appl. S. 13 (2003) 1673-1687. | Zbl 1056.65108

[9] D. Boffi, M. Costabel, M. Dauge and L. Demkowicz, Discrete compactness for the hp version of rectangular edge finite elements. ICES Report 04-29, University of Texas, Austin (2004). | Zbl 1122.65110

[10] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer, New York (1991). | MR 1115205 | Zbl 0788.73002

[11] F. Chatelin, Spectral approximation of linear operators. Academic Press, New York (1983). | MR 716134 | Zbl 0517.65036

[12] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[13] M. Costabel and M. Dauge, Singularities of electromagnetic fields in polyhedral domains. Arch. Rational Mech. Anal. 151 (2000) 221-276. | Zbl 0968.35113

[14] M. Dauge, Benchmark computations for Maxwell equations for the approximation of highly singular solutions. Technical report, University of Rennes 1. http://perso.univ-rennes1.fr/monique.dauge/core/index.html

[15] L. Demkowicz, P. Monk, C. Schwab and L. Vardepetyan, Maxwell eigenvalues and discrete compactness in two dimensions. Comput. Math. Appl. 40 (2000) 589-605. | Zbl 0998.78011

[16] C. Hazard and M. Lenoir, On the solution of time-harmonic scattering problems for Maxwell's equations. SIAM J. Math. Anal. 27 (1996) 1597-1630. | Zbl 0860.35129

[17] J. Hesthaven and T. Warburton, High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem. Philos. T. Roy. Soc. A 362 (2004) 493-524. | Zbl 1078.78014

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

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

[20] P. Houston, I. Perugia and D. Schötzau, Mixed discontinuous Galerkin approximation of the Maxwell operator: Non-stabilized formulation. J. Sci. Comput. 22 (2005) 315-346. | Zbl 1091.78017

[21] F. Kikuchi, On a discrete compactness property for the Nédélec finite elements. J. Fac. Sci. U. Tokyo IA 36 (1989) 479-490. | Zbl 0698.65067

[22] M. Krizek and P. Neittaanmaki, On the validity of Friedrichs' inequalities. Math. Scand. 54 (1984) 17-26. | Zbl 0555.35003

[23] R. Leis, Initial boundary value problems in Mathematical Physics. John Wiley, New York (1988). | Zbl 0599.35001

[24] S. Lohrengel and S. Nicaise, A discontinuous Galerkin method on refined meshes for the 2d time-harmonic Maxwell equations in composite materials. Preprint Macs, University of Valenciennes, 2004. J. Comput. Appl. Math. (to appear). | MR 2333834 | Zbl 1141.65080

[25] P. Monk, Finite element methods for Maxwell's equations. Numer. Math. Scientific Comp., Oxford Univ. Press, New York (2003). | Zbl 1024.78009

[26] P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell’s equations in 3 . Math. Comp. 70 (2000) 507-523. | Zbl 1035.65131

[27] J. Osborn, Spectral approximation for compact operators. Math. Comp. 29 (1975) 712-725. | Zbl 0315.35068