Substructuring preconditioners for saddle-point problems arising from Maxwell's equations in three dimensions

Mathematics of Computation

Volumn 73, Issue 245, 2004, Pages 35-61

  Hu, Qiya ^a   Zou, Jun ^b  

^a INSTITUTE OF COMPUTATIONAL MATHEMATICS AND SCIENTIFIC ENGINEERING COMPUTING   (China)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Condition numbers; Maxwell's equations; N d lec finite elements; Nonoverlapping domain decomposition

Indexed keywords

EID: 0942289764     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01541-2     Document Type: Article

References (32)

1. A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of H(curl;Ω) and the construction of an extension operator, Manuscr. Math., 89(1986), 159-178. MR 96k:46057
2. A. Alonso and A. Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp., 68(1999), 607-631. MR 99i:78002
3. C. Amrouche, C. Bernardi, M. Dauge and V. Girault. Vector Potentials in three dimensional nonsmooth domains, Math. Meth. Appl. Sci., 21(1998), 823-864. MR 99e:35037
4. J. Bramble, J. Pasciak and A. Schatz, The construction of preconditioner for elliptic problems by substructuring, IV. Math. Comp., 53(1989), 1-24. MR 89m:65098
5. J. Bramble, J. Pasciak and A. Vassilev, Analysis of the inexact Uzawa algorithm for saddle-point problems, SIAM J. Numer. Anal., 34(1997), 1072-1092. MR 98c:65182
6. J. Bramble and J. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp., 50(1988), 1-18. MR 89m:65097a
7. M. Cessenat, Mathematical methods in electromagnetism, World Scientific, River Edge, NJ, 1998. MR 97j:78001
8. Z. Chen, Q. Du and J. Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal., 37(1999), 1542-1570. MR 2001h:78044
9. P. Ciarlet, Jr. and J. Zou, Fully discrete finite element approaches for time-dependent Maxwell's equations, Numer. Math., 82(1999), 193-219 MR 2000c:65083
10. M. Costabel, A remark on the regularity of solution of Maxwell's equations on Lipschitz domains, Math. Meth. Appl. Sci. 12 (1990), 365-368. MR 91c:35028
11. R. Dautray and J.-L. Lions, Mathematical analysis and numerical methods for science and technology, Springer-Verlag, New York, 1988. MR 89m:00001
12. V. Girault and P. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. MR 88b:65129
13. J. Gopalakrishnan and J. Pasciak, Overlapping Schwarz preconditioned for indefinite time harmonic Maxwell's equations, Math. Comp., 72 (2003), 1-16.
14. R. Hiptmair, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal., 36(1998), 204-225. MR 99j:65229
15. Q. Hu and G. Liang, A general framework to construct interface preconditioned, Chinese J. Num. Math. & Appl., 21(1999), 83-95. MR 99m:65238
16. Q. Hu, G. Liang and J. Lui, Construction of a preconditioner for three dimensional domain decomposition methods with Lagrangian multipliers, J. Comput. Math., 19 (2001), 213-224. MR 2001k:65186
17. Q. Hu and J. Zou, An iterative method with variable parameters for saddle-point problems, SIAM J. Matrix Anal. Appl., 23(2001), 317-338. MR 2002j:65041
18. Q. Hu and J. Zou, Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems, Numer. Math., 93 (2002), 333-359.
19. Q. Hu and J. Zou, Substructuring preconditioned for saddle-point problems arising from Maxwell's equations in three dimensions. Technical Report CUHK 2002-16(256), Department of Mathematics, The Chinese University of Hong Kong, 2002.
20. Q. Hu and J. Zou, A non-overlapping domain decomposition method for Maxwell's equations in three dimensions. Accepted for publication in SIAM J. Numer. Anal.
21. P. Monk, Analysis of a finite element method for Maxwell's equations, SIAM J. Numer. Anal., 29(1992), 32-56. MR 93k:65096
22. 3, Numer. Math., 35(1980), 315-341. MR 81k:65125
23. R. Nicolaides and D. Wang, Convergence analysis of a covolume scheme for Maxwell's equations in three dimensions, Math. Comp., 67(1998), 947-963. MR 98j:65080
24. T. Rusten and R. Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl., 13(1992), 887-904. MR 93a:65043
25. J. Saranen, On electric and magnetic problems for vector fields in anisotropic nonhomogeneous media, J. Math. Anal. Appl. 91(1983), 254-275. MR 851:78004
26. B. Smith, A domain decomposition algorithm for elliptic problems in three dimensions, Numer. Math., 60(1991), 219-234. MR 92m:65159
27. B. Smith, P. Bjorstad and W. Gropp, Domain Decomposition: Parallel multilevel methods for elliptic partial differential equations, Cambridge University Press, 1996. MR 98g:65003
28. P. Tallec, Domain Decomposition Methods in Computational Mechanics, Comput. Mech. Adv., 2: 1321-220, 1994.
29. A. Toselli, Overlapping Schwarz methods for Maxwell's equations in three dimensions, Numer. Math., 86(2000), 733-752. MR 2001h:65137
30. A. Toselli, O. Widlund and B. Wohlmuth, An iterative substructuring method for Maxwell's equations in two dimensions, Math. Comp. 70 (2001), 935-947. MR 2001j:65140
31. J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Review, 34(1992), 581-613. MR 93k:65029
32. J. Xu and J. Zou, Some nonoverlapping domain decomposition methods, SIAM Review, 24(1998), 857-914. MR 99m:65241