Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations

Authors: Jayadeep Gopalakrishnan and Joseph E. Pasciak
Journal: Math. Comp. 72 (2003), 1-15
MSC (2000): Primary 65F10, 65N55, 65N30
Published electronically: December 5, 2001
MathSciNet review: 1933811
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Time harmonic Maxwell equations in lossless media lead to a second order differential equation for the electric field involving a differential operator that is neither elliptic nor definite. A Galerkin method using Nedelec spaces can be employed to get approximate solutions numerically. The problem of preconditioning the indefinite matrix arising from this method is discussed here. Specifically, two overlapping Schwarz methods will be shown to yield uniform preconditioners.

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

  • 1. C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional nonsmooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823-864. MR 99e:35037
  • 2. Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Multigrid in $\mathbf{H}(\mathrm{div})$ and $\mathbf{H}(\mathbf{curl})$, Numer. Math. 85 (2000), no. 2, 197-217. MR 2001d:65161
  • 3. Rudolf Beck, Peter Deuflhard, Ralf Hiptmair, Ronald H. W. Hoppe, and Barbara Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, Surveys Math. Indust. 8 (1999), 271-312. MR 2000i:65206
  • 4. Alain Bossavit, A rationale for ``edge-elements'' in 3-D fields computations, IEEE Trans. Mag. 24 (1988), no. 1, 74-79.
  • 5. Franco Brezzi and Michel Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, no. 15, Springer-Verlag, New York, 1991. MR 92d:65187
  • 6. Xiao-Chuan Cai and Olof B. Widlund, Domain decomposition algorithms for indefinite elliptic problems, SIAM J. Sci. Stat. Comput. 13 (1992), no. 1, 243-258. MR 92i:65181
  • 7. L. Demkowicz and L. Vardapetyan, Modeling of electromagnetic absorption/scattering problems using $hp$-adaptive finite elements, Comput. Methods Appl. Mech. Engrg. 152 (1998), 103-124. MR 99b:78003 20pt
  • 8. M. Dryja and O. B. Widlund, An additive variant of the Schwarz alternating method for the case of many subregions, Tech. Report 339, Courant Institute of Mathematical Sciences, New York, 1987.
  • 9. -, Some domain decompostion algorithms for elliptic problems, Iterative Methods for Large Linear Systems, Academic Press, San Diego, 1989, Proceedings of a conference held at Austin, Texas, in October 1988, pp. 273-291. MR 91f:65071
  • 10. -, Towards a unified theory of domain decomposition algorithms for elliptic problems, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1990, Symposium held at Houston, Texas, in March 1989, 3-21. MR 91m:65294
  • 11. Stanley C. Eisenstat, Howard C. Elman, and Martin H. Schultz, Variational iterative methods for nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 20 (1983), no. 2, 345-357. MR 84h:65030
  • 12. V. Girault, Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in ${\mathbf R}^3$, Math. Comp. 51 (1988), no. 183, 55-74. MR 90e:65155
  • 13. Vivette Girault and Pierre-Arnaud Raviart, Finite Element Methods for Navier-Stokes Equations, Springer series in Computational Mathematics, no. 5, Springer-Verlag, New York, 1986. MR 88b:65129
  • 14. Jayadeep Gopalakrishnan, Joseph E. Pasciak, and Leszek Demkowicz, A multigrid algorithm for time harmonic Maxwell equations, In preparation.
  • 15. Ralf Hiptmair, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 204-225. MR 99j:65229
  • 16. Ralf Hiptmair and Andrea Toselli, Overlapping Schwarz methods for vector valued elliptic problems in three dimensions, Parallel solution of PDEs, IMA Volumes in Mathematics and its Applications, Springer-Verlag, Berlin, 1998.
  • 17. R. Leis, Exterior boundary-value problems in mathematical physics, Trends in Applications of Pure Mathematics to Mechanics, Volume II (Henryk Zorski, ed.), Monographs and Studies in Mathematics, no. 5, Pitman, London, 1979, A collection of papers presented at a symposium at Kozubnik, Poland, in September 1977, pp. 187-203. MR 81d:78016
  • 18. P. L. Lions, On the Schwarz alternating method, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Roland Glowinski, Gene H. Golub, Gérard A. Meurant, and Jacques Périaux, eds.), SIAM, Philadelphia, 1988, Symposium held at Ecole Nationale des Ponts et Chaussées, Paris, in January 1987, pp. 1-42. MR 90a:65248
  • 19. P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell's equations in ${\bf {R}}\sp 3$, Math. Comp. 70 (2001), 507-523. MR 2001g:65156
  • 20. Peter Monk, A finite element method for aproximating the time-harmonic Maxwell equations, Numer. Math. 63 (1992), 243-261. MR 94b:65134
  • 21. J. C. Nedelec, Mixed Finite Elements in ${\mathbb R}^3$, Numer. Math. 35 (1980), 315-341. MR 81k:65125
  • 22. Waldemar Rachowicz, Leszek Demkowicz, Andrzej Bajer, and Timothy Walsh, A two-grid iterative solver for stationary Maxwell's equations, Iterative Methods in Scientific Computation II (D. Kincaid et al., eds.), IMACS, 1999.
  • 23. Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986), 856-869. MR 87g:65064
  • 24. Barry F. Smith, Petter E. Bjørstad, and William D. Gropp, Domain Decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, 1996. MR 98g:65003
  • 25. A. Toselli, Overlapping Schwarz methods for Maxwell's equations in three dimensions, Numer. Math. 86 (2000), 733-752. MR 2001h:65137
  • 26. Andrea Toselli, Olof B. Widlund, and Barbara I. Wohlmuth, An iterative substructuring method for Maxwell's equations in two dimensions, Math. Comp. 70 (2000), 935-949. MR 2001j:65140

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65F10, 65N55, 65N30

Retrieve articles in all journals with MSC (2000): 65F10, 65N55, 65N30

Additional Information

Jayadeep Gopalakrishnan
Affiliation: Institute for Mathematics and its Applications, Minneapolis, Minnesota 55455

Joseph E. Pasciak
Affiliation: Texas A&M University, College Station, Texas 77843-3368.

Keywords: Schwarz method, indefinite, Maxwell equations, preconditioner, domain decomposition, finite element
Received by editor(s): July 10, 2000
Received by editor(s) in revised form: March 7, 2001
Published electronically: December 5, 2001
Additional Notes: The first author was supported in part by Medtronic Inc
The second author was partially supported by NSF grant number DMS-9973328
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society