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Computing electromagnetic eigenmodes with continuous Galerkin approximations. (English) Zbl 1194.78053
Summary: Costabel and Dauge proposed a variational setting to solve numerically the time-harmonic Maxwell equations in 3D polyhedral geometries, with a continuous approximation of the electromagnetic field. In order to remove spurious eigenmodes, three computational strategies are then possible. The original method, which requires a parameterization of the variational formulation. The second method, which is based on an a posteriori filtering of the computed eigenmodes. And the third method, which uses a mixed variational setting so that all spurious modes are removed a priori. In this paper, we discuss the relative merits of the approaches, which are illustrated by a series of 3D numerical examples.
MSC:
78M10Finite element methods (optics)
78A25General electromagnetic theory
References:
[1]Amrouche, C.; Bernardi, C.; Dauge, M.; Girault, V.: Vector potentials in three-dimensional non-smooth domains, Math. methods appl. Sci. 21, 823-864 (1998) · Zbl 0914.35094 · doi:10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
[2]Assous, F.; Degond, P.; Heintzé, E.; Raviart, P. -A.; Segré, J.: On a finite element method for solving the three-dimensional Maxwell equations, J. comput. Phys. 109, 222-237 (1993) · Zbl 0795.65087 · doi:10.1006/jcph.1993.1214
[3]Babuska, I.; Osborn, J. E.: Eigenvalue problems, Handbook of numerical analysis, 641-787 (1991) · Zbl 0875.65087
[4]Boffi, D.: Three-dimensional finite element methods for the Stokes problem, SIAM J. Numer. anal. 34, 664-670 (1997) · Zbl 0874.76032 · doi:10.1137/S0036142994270193
[5]Boffi, D.: Compatible discretizations for eigenvalue problems, IMA volumes in mathematics and its applications 142, 121-142 (2006) · Zbl 1110.65104
[6]Boffi, D.; Brezzi, F.; Gastaldi, L.: On the convergence of eigenvalues for mixed formulations, Ann. sci. Norm. sup. Pisa cl. Sci. 25, 131-154 (1997) · Zbl 1003.65052 · doi:numdam:ASNSP_1997_4_25_1-2_131_0
[7]Boffi, D.; Costabel, M.; Dauge, M.; Demkowicz, L.: Discrete compactness for the hp version of rectangular edge finite elements, SIAM J. Numer. anal. 44, 979-1004 (2006) · Zbl 1122.65110 · doi:10.1137/04061550X
[8]Bossavit, A.; Rapetti, F.: Geometrical localisation of the degrees of freedom for Whitney elements of higher order, IET sci. Meas. technol. 1, 63-66 (2007)
[9]Brezzi, F.; Fortin, M.: Mixed and hybrid finite element methods, Springer series in computational mathematics 15 (1991) · Zbl 0788.73002
[10]A. Buffa, P. Ciarlet Jr., E. Jamelot, Solving electromagnetic eigenvalue problems in polyhedral domains. Numer. Math., submitted for publication.
[11]Jr., P. Ciarlet: Augmented formulations for solving Maxwell equations, Comput. methods appl. Mech. engrg. 194, 559-586 (2005) · Zbl 1063.78018 · doi:10.1016/j.cma.2004.05.021
[12]Jr., P. Ciarlet; Girault, V.: Inf – sup condition for the 3D, P2-iso-P1 Taylor – Hood finite element; application to Maxwell equations, CR acad. Sci. Paris, ser. I. 335, 827-832 (2002) · Zbl 1021.78009 · doi:10.1016/S1631-073X(02)02564-5
[13]P. Ciarlet Jr., G. Hechme, Mixed, augmented variational formulations for Maxwell’s equations: numerical analysis via the macroelement technique. Numer. Math., submitted for publication.
[14]P. Ciarlet Jr., F. Lefèvre, S. Lohrengel, S. Nicaise, Weighted regularization for composite materials in electromagnetism. Math. Mod. Numer. Anal., submitted for publication.
[15]Costabel, M.: A coercive bilinear form for Maxwell’s equations, J. math. Anal. appl. 157, 527-541 (1991) · Zbl 0738.35095 · doi:10.1016/0022-247X(91)90104-8
[16]Costabel, M.; Dauge, M.: Singularities of Maxwell’s equations on polyhedral domains, Pitman research notes in mathematics series 379, 69-76 (1998) · Zbl 0904.35089
[17]Costabel, M.; Dauge, M.: Weighted regularization of Maxwell equations in polyhedral domains, Numer. math. 93, 239-277 (2002) · Zbl 1019.78009 · doi:10.1007/s002110100388
[18]Costabel, M.; Dauge, M.: Computation of resonance frequencies for Maxwell equations in non smooth domains, Lecture notes in computer science and engineering 31 (2003) · Zbl 1116.78002
[19]M. Dauge, Benchmark computations for Maxwell equations for the approximation of highly singular solutions (2004). See Monique Dauge’s personal web page at the location. lt;http://perso.univ-rennes1.fr/monique.dauge/core/index.htmlgt;.
[20]E. Heintzé, Solution to the 3D instationary Maxwell equations with conforming finite elements (in French). PhD Thesis, Université Paris VI, France, 1992.
[21]Kikuchi, F.: Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, Comput. methods appl. Mech. engrg. 64, 509-521 (1987) · Zbl 0644.65087 · doi:10.1016/0045-7825(87)90053-3
[22]Monk, P.; Demkowicz, L.: Discrete compactness and the approximation of Maxwell’s equations in R3, Math. comput. 70, 507-523 (2001) · Zbl 1035.65131 · doi:10.1090/S0025-5718-00-01229-1
[23]Nédélec, J. -C.: Mixed finite elements in R3, Numer. math. 35, 315-341 (1980) · Zbl 0419.65069 · doi:10.1007/BF01396415
[24]Stenberg, R.: Analysis of mixed finite element methods for the Stokes problem: a unified approach, Math. comput. 42, 9-23 (1984) · Zbl 0535.76037 · doi:10.2307/2007557
[25]Weber, C.: A local compactness theorem for Maxwell’s equations, Math. methods appl. Sci. 2, 12-25 (1980) · Zbl 0432.35032 · doi:10.1002/mma.1670020103