Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



An optimal domain decomposition preconditioner
for low-frequency time-harmonic
Maxwell equations

Authors: Ana Alonso and Alberto Valli
Journal: Math. Comp. 68 (1999), 607-631
MSC (1991): Primary 65N55, 65N30; Secondary 35Q60
MathSciNet review: 1609607
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

References [Enhancements On Off] (What's this?)

  • 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. MR 0450957
  • 2. Ana Alonso and Alberto Valli, Some remarks on the characterization of the space of tangential traces of 𝐻(𝑟𝑜𝑡;Ω) and the construction of an extension operator, Manuscripta Math. 89 (1996), no. 2, 159–178. MR 1371994, 10.1007/BF02567511
  • 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. MR 1442391, 10.1016/S0045-7825(96)01144-9
  • 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.
  • 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.
  • 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. MR 1048563, 10.1002/mma.1670120406
  • 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. MR 961439
  • 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). MR 1026513
  • 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. MR 566529
  • 10. Peter Monk, A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math. 63 (1992), no. 2, 243–261. MR 1182977, 10.1007/BF01385860
  • 11. Peter Monk, Analysis of a finite element method for Maxwell’s equations, SIAM J. Numer. Anal. 29 (1992), no. 3, 714–729. MR 1163353, 10.1137/0729045
  • 12. J.-C. Nédélec, Mixed finite elements in 𝑅³, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, 10.1007/BF01396415
  • 13. J.-C. Nédélec, A new family of mixed finite elements in 𝑅³, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, 10.1007/BF01389668
  • 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. MR 1128036, 10.1007/BF01385813
  • 15. Jukka Saranen, On generalized harmonic fields in domains with anisotropic nonhomogeneous media, J. Math. Anal. Appl. 88 (1982), no. 1, 104–115. MR 661405, 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. MR 688547, 10.1016/0022-247X(83)90107-5
  • 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. MR 688544, 10.1016/0022-247X(83)90104-X
  • 17. A. Valli, Orthogonal decompositions of $(L^2(\Omega ))^3$, preprint UTM 493, Dipartimento di Matematica, Università di Trento, 1996.

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65N55, 65N30, 35Q60

Retrieve articles in all journals with MSC (1991): 65N55, 65N30, 35Q60

Additional Information

Ana Alonso
Affiliation: Dipartimento di Matematica, Università di Trento, 38050 Povo (Trento), Italy

Alberto Valli
Affiliation: Dipartimento di Matematica, Università di Trento, 38050 Povo (Trento), Italy

Keywords: Domain decomposition methods, Maxwell equations
Received by editor(s): December 2, 1996
Received by editor(s) in revised form: July 30, 1997
Additional Notes: Partially supported by H.C.M. contract CHRX 0930407
Article copyright: © Copyright 1999 American Mathematical Society