Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations

Gopalakrishnan, Jayadeep; Pasciak, Joseph

doi:10.1090/S0025-5718-01-01406-5

Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations
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by Jayadeep Gopalakrishnan and Joseph E. Pasciak PDF

Math. Comp. 72 (2003), 1-15 Request permission

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

C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823–864 (English, with English and French summaries). MR 1626990, DOI 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Multigrid in $H(\textrm {div})$ and $H(\textrm {curl})$, Numer. Math. 85 (2000), no. 2, 197–217. MR 1754719, DOI 10.1007/PL00005386
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), no. 3-4, 271–312. MR 1737416
Alain Bossavit, A rationale for “edge-elements” in 3–D fields computations, IEEE Trans. Mag. 24 (1988), no. 1, 74–79.
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Xiao-Chuan Cai and Olof B. Widlund, Domain decomposition algorithms for indefinite elliptic problems, SIAM J. Sci. Statist. Comput. 13 (1992), no. 1, 243–258. MR 1145185, DOI 10.1137/0913013
Lawrence M. Graves, The Weierstrass condition for multiple integral variation problems, Duke Math. J. 5 (1939), 656–660. MR 99
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.
David R. Kincaid and Linda J. Hayes (eds.), Iterative methods for large linear systems, Academic Press, Inc., Boston, MA, 1990. Papers from the conference held at the University of Texas at Austin, Austin, Texas, October 19–21, 1988. MR 1038083
Maksymilian Dryja and Olof B. Widlund, Towards a unified theory of domain decomposition algorithms for elliptic problems, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) SIAM, Philadelphia, PA, 1990, pp. 3–21. MR 1064335
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 694523, DOI 10.1137/0720023
V. Girault, Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in $\textbf {R}^3$, Math. Comp. 51 (1988), no. 183, 55–74. MR 942143, DOI 10.1090/S0025-5718-1988-0942143-2
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
Jayadeep Gopalakrishnan, Joseph E. Pasciak, and Leszek Demkowicz, A multigrid algorithm for time harmonic Maxwell equations, In preparation.
R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 204–225. MR 1654571, DOI 10.1137/S0036142997326203
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.
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
P.-L. Lions, On the Schwarz alternating method. I, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 1–42. MR 972510
P. Monk and L. Demkowicz, Discrete compactness and the approximation of Maxwell’s equations in ${\Bbb R}^3$, Math. Comp. 70 (2001), no. 234, 507–523. MR 1709155, DOI 10.1090/S0025-5718-00-01229-1
Peter Monk, A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math. 63 (1992), no. 2, 243–261. MR 1182977, DOI 10.1007/BF01385860
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
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.
Youcef Saad and Martin H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986), no. 3, 856–869. MR 848568, DOI 10.1137/0907058
Barry F. Smith, Petter E. Bjørstad, and William D. Gropp, Domain decomposition, Cambridge University Press, Cambridge, 1996. Parallel multilevel methods for elliptic partial differential equations. MR 1410757
Andrea Toselli, Overlapping Schwarz methods for Maxwell’s equations in three dimensions, Numer. Math. 86 (2000), no. 4, 733–752. MR 1794350, DOI 10.1007/PL00005417
Andrea Toselli, Olof B. Widlund, and Barbara I. Wohlmuth, An iterative substructuring method for Maxwell’s equations in two dimensions, Math. Comp. 70 (2001), no. 235, 935–949. MR 1710632, DOI 10.1090/S0025-5718-00-01244-8