# zbMATH — the first resource for mathematics

An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations. (English) Zbl 1043.78554
Summary: The time-harmonic Maxwell equations are considered in the low-frequency case. A finite element domain decomposition approach is proposed for the numerical approximation of the exact solution. This leads to an iteration-by-subdomain procedure, which is proven to converge. The rate of convergence turns out to be independent of the mesh size, showing that the preconditioner implicitly defined by the iterative procedure is optimal. For obtaining this convergence result it has been necessary to prove a regularity theorem for Dirichlet and Neumann harmonic fields.

##### MSC:
 78M25 Numerical methods in optics (MSC2010) 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35Q60 PDEs in connection with optics and electromagnetic theory
##### Keywords:
domain decomposition methods; Maxwell equations
Full Text:
##### References:
 [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030 [2] Ana Alonso and Alberto Valli, Some remarks on the characterization of the space of tangential traces of \?(\?\?\?;\Omega ) and the construction of an extension operator, Manuscripta Math. 89 (1996), no. 2, 159 – 178. · Zbl 0856.46019 [3] Ana Alonso and Alberto Valli, A domain decomposition approach for heterogeneous time-harmonic Maxwell equations, Comput. Methods Appl. Mech. Engrg. 143 (1997), no. 1-2, 97 – 112. · Zbl 0883.65096 [4] A. Alonso and A. Valli, Unique solvability for high-frequency heterogeneous time-harmonic Maxwell equations via the Fredholm alternative theory, Math. Meth. Appl. Sci., to appear. · Zbl 0923.35178 [5] C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains, preprint R 96001, Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, Paris, 1996. · Zbl 0914.35094 [6] Martin Costabel, A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains, Math. Methods Appl. Sci. 12 (1990), no. 4, 365 – 368. · Zbl 0699.35028 [7] Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. · Zbl 0668.35001 [8] Michal Křížek and Pekka Neittaanmäki, On time-harmonic Maxwell equations with nonhomogeneous conductivities: solvability and FE-approximation, Apl. Mat. 34 (1989), no. 6, 480 – 499 (English, with Russian and Czech summaries). · Zbl 0696.65085 [9] R. Leis, Exterior boundary-value problems in mathematical physics, Trends in applications of pure mathematics to mechanics, Vol. II (Second Sympos., Kozubnik, 1977) Monographs Stud. Math., vol. 5, Pitman, Boston, Mass.-London, 1979, pp. 187 – 203. · Zbl 0414.73082 [10] Peter Monk, A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math. 63 (1992), no. 2, 243 – 261. · Zbl 0757.65126 [11] Peter Monk, Analysis of a finite element method for Maxwell’s equations, SIAM J. Numer. Anal. 29 (1992), no. 3, 714 – 729. · Zbl 0761.65097 [12] J.-C. Nédélec, Mixed finite elements in \?³, Numer. Math. 35 (1980), no. 3, 315 – 341. · Zbl 0419.65069 [13] J.-C. Nédélec, A new family of mixed finite elements in \?³, Numer. Math. 50 (1986), no. 1, 57 – 81. · Zbl 0625.65107 [14] A. Quarteroni, G. Sacchi Landriani, and A. Valli, Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements, Numer. Math. 59 (1991), no. 8, 831 – 859. · Zbl 0738.76044 [15] Jukka Saranen, On generalized harmonic fields in domains with anisotropic nonhomogeneous media, J. Math. Anal. Appl. 88 (1982), no. 1, 104 – 115. , https://doi.org/10.1016/0022-247X(82)90179-2 Jukka Saranen, Erratum: ”On generalized harmonic fields in domains with anisotropic nonhomogeneous media”, J. Math. Anal. Appl. 91 (1983), no. 1, 300. , https://doi.org/10.1016/0022-247X(83)90107-5 Jukka Saranen, On generalized harmonic fields in domains with anisotropic nonhomogeneous media, J. Math. Anal. Appl. 88 (1982), no. 1, 104 – 115. , https://doi.org/10.1016/0022-247X(82)90179-2 Jukka Saranen, Erratum: ”On generalized harmonic fields in domains with anisotropic nonhomogeneous media”, J. Math. Anal. Appl. 91 (1983), no. 1, 300. · Zbl 0521.35008 [16] Jukka Saranen, On electric and magnetic problems for vector fields in anisotropic nonhomogeneous media, J. Math. Anal. Appl. 91 (1983), no. 1, 254 – 275. · Zbl 0519.35010 [17] A. Valli, Orthogonal decompositions of $$(L^2(\Omega ))^3$$, preprint UTM 493, Dipartimento di Matematica, Università di Trento, 1996.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.