Preconditioners for the discretized time-harmonic Maxwell equations in mixed form. (English) Zbl 1199.78010

Summary: We introduce a new preconditioning technique for iteratively solving linear systems arising from finite element discretization of the mixed formulation of time-harmonic Maxwell equations. The preconditioners are motivated by spectral equivalence properties of discrete operators, but are augmentation free and Schur complement free. We provide a complete spectral analysis, and show that the eigenvalues of the preconditioned saddle point matrix are strongly clustered. The analytical observations are accompanied by numerical results that demonstrate the scalability of the proposed approach.


78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
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[1] Chen, SIAM Journal on Numerical Analysis 37 pp 1542– (2000)
[2] Demkowicz, Computer Methods in Applied Mechanics and Engineering 152 pp 103– (1998)
[3] Houston, Modélisation Mathematique et Analyse Numérique 39 pp 727– (2005)
[4] Perugia, Computer Methods in Applied Mechanics and Engineering 191 pp 4675– (2002)
[5] Nédélec, Numerische Mathematik 35 pp 315– (1980)
[6] Finite Element Methods for Maxwell’s Equations. Oxford University Press: New York, 2003. · Zbl 1024.78009
[7] Hiptmair, Acta Numerica 11 pp 237– (2002)
[8] Haber, Journal of Computational Physics 163 pp 150– (2000)
[9] Gopalakrishnan, SIAM Journal on Numerical Analysis 42 pp 90– (2004)
[10] Reitzinger, Numerical Linear Algebra with Applications 9 pp 223– (2002)
[11] Benzi, Acta Numerica 14 pp 1– (2005)
[12] Hu, Mathematics of Computation 73 pp 35– (2004)
[13] Arnold, Numerische Mathematik 85 pp 197– (2000)
[14] Hiptmair, SIAM Journal on Numerical Analysis 36 pp 204– (1998)
[15] Hu, SIAM Journal on Scientific Computing 27 pp 1669– (2006)
[16] . Mixed and hybrid finite element methods. In Springer Series in Computational Mathematics, vol. 15. Springer: New York, 1991.
[17] , . Iterative methods for problems in computational fluid dynamics. In Iterative Methods in Scientific Computing, , (eds). Springer: Singapore, 1997; 271–327. · Zbl 0957.76071
[18] , . Finite Elements and Fast Iterative Solvers. Oxford University Press: Oxford, 2005.
[19] Greif, ETNA, Special Volume on Saddle Point Problems 22 pp 114– (2006)
[20] Iterative Methods for Solving Linear Systems. SIAM: Philadelphia, MA, 1997.
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