Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions
HTML articles powered by AMS MathViewer
- by Qiya Hu and Jun Zou;
- Math. Comp. 73 (2004), 35-61
- DOI: https://doi.org/10.1090/S0025-5718-03-01541-2
- Published electronically: August 19, 2003
- PDF | Request permission
Abstract:
This paper is concerned with the saddle-point problems arising from edge element discretizations of Maxwell’s equations in a general three dimensional nonconvex polyhedral domain. A new augmented technique is first introduced to transform the problems into equivalent augmented saddle-point systems so that they can be solved by some existing preconditioned iterative methods. Then some substructuring preconditioners are proposed, with very simple coarse solvers, for the augmented saddle-point systems. With the preconditioners, the condition numbers of the preconditioned systems are nearly optimal; namely, they grow only as the logarithm of the ratio between the subdomain diameter and the finite element mesh size.References
- Ana Alonso and Alberto Valli, Some remarks on the characterization of the space of tangential traces of $H(\textrm {rot};\Omega )$ and the construction of an extension operator, Manuscripta Math. 89 (1996), no. 2, 159–178. MR 1371994, DOI 10.1007/BF02567511
- Ana Alonso and Alberto Valli, An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations, Math. Comp. 68 (1999), no. 226, 607–631. MR 1609607, DOI 10.1090/S0025-5718-99-01013-3
- 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
- James H. Bramble, Joseph E. Pasciak, and Alfred H. Schatz, The construction of preconditioners for elliptic problems by substructuring. IV, Math. Comp. 53 (1989), no. 187, 1–24. MR 970699, DOI 10.1090/S0025-5718-1989-0970699-3
- James H. Bramble, Joseph E. Pasciak, and Apostol T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997), no. 3, 1072–1092. MR 1451114, DOI 10.1137/S0036142994273343
- James H. Bramble and Joseph E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comp. 50 (1988), no. 181, 1–17. MR 917816, DOI 10.1090/S0025-5718-1988-0917816-8
- Michel Cessenat, Mathematical methods in electromagnetism, Series on Advances in Mathematics for Applied Sciences, vol. 41, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. Linear theory and applications. MR 1409140, DOI 10.1142/2938
- Zhiming Chen, Qiang Du, and Jun Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. Anal. 37 (2000), no. 5, 1542–1570. MR 1759906, DOI 10.1137/S0036142998349977
- P. Ciarlet Jr. and Jun Zou, Fully discrete finite element approaches for time-dependent Maxwell’s equations, Numer. Math. 82 (1999), no. 2, 193–219 (English, with English and French summaries). MR 1685459, DOI 10.1007/s002110050417
- 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, DOI 10.1002/mma.1670120406
- Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 2, Springer-Verlag, Berlin, 1988. Functional and variational methods; With the collaboration of Michel Artola, Marc Authier, Philippe Bénilan, Michel Cessenat, Jean Michel Combes, Hélène Lanchon, Bertrand Mercier, Claude Wild and Claude Zuily; Translated from the French by Ian N. Sneddon. MR 969367, DOI 10.1007/978-3-642-61566-5
- 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
- J. Gopalakrishnan and J. Pasciak, Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell’s equations, Math. Comp., 72 (2003), 1-16.
- R. Hiptmair, Multigrid method for Maxwell’s equations, SIAM J. Numer. Anal. 36 (1999), no. 1, 204–225. MR 1654571, DOI 10.1137/S0036142997326203
- Qi Ya Hu and Guo Ping Liang, A general framework for constructing an interface preconditioner for domain decomposition methods, Math. Numer. Sin. 21 (1999), no. 1, 117–128 (Chinese, with English summary); English transl., Chinese J. Numer. Math. Appl. 21 (1999), no. 2, 83–94. MR 1676186
- Qi-ya Hu, Guo-ping Liang, and Jin-zhao Liu, Construction of a preconditioner for domain decomposition methods with polynomial Lagrangian multipliers, J. Comput. Math. 19 (2001), no. 2, 213–224. MR 1816686
- Qiya Hu and Jun Zou, An iterative method with variable relaxation parameters for saddle-point problems, SIAM J. Matrix Anal. Appl. 23 (2001), no. 2, 317–338. MR 1871315, DOI 10.1137/S0895479899364064
- Q. Hu and J. Zou, Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems, Numer. Math., 93 (2002), 333–359.
- Q. Hu and J. Zou, Substructuring preconditioners 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.
- 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.
- Peter Monk, Analysis of a finite element method for Maxwell’s equations, SIAM J. Numer. Anal. 29 (1992), no. 3, 714–729. MR 1163353, DOI 10.1137/0729045
- J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
- R. A. Nicolaides and D-Q. Wang, Convergence analysis of a covolume scheme for Maxwell’s equations in three dimensions, Math. Comp. 67 (1998), no. 223, 947–963. MR 1474654, DOI 10.1090/S0025-5718-98-00971-5
- Torgeir Rusten and Ragnar Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl. 13 (1992), no. 3, 887–904. Iterative methods in numerical linear algebra (Copper Mountain, CO, 1990). MR 1168084, DOI 10.1137/0613054
- 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, DOI 10.1016/0022-247X(83)90104-X
- Barry F. Smith, A domain decomposition algorithm for elliptic problems in three dimensions, Numer. Math. 60 (1991), no. 2, 219–234. MR 1133580, DOI 10.1007/BF01385722
- 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
- P. Tallec, Domain Decomposition Methods in Computational Mechanics, Comput. Mech. Adv., 2: 1321-220, 1994.
- 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
- Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, DOI 10.1137/1034116
- Jinchao Xu and Jun Zou, Some nonoverlapping domain decomposition methods, SIAM Rev. 40 (1998), no. 4, 857–914. MR 1659681, DOI 10.1137/S0036144596306800
Bibliographic Information
- Qiya Hu
- Affiliation: Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China
- Email: hqy@lsec.cc.ac.cn
- Jun Zou
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
- ORCID: 0000-0002-4809-7724
- Email: zou@math.cuhk.edu.hk
- Received by editor(s): February 21, 2002
- Received by editor(s) in revised form: July 15, 2002
- Published electronically: August 19, 2003
- Additional Notes: The work of the first author was supported by Special Funds for Major State Basic Research Projects of China G1999032804.
The work of the second author was partially supported by Hong Kong RGC Grants CUHK4048/02P and CUHK4292/00P - © Copyright 2003 American Mathematical Society
- Journal: Math. Comp. 73 (2004), 35-61
- MSC (2000): Primary 65N30, 65N55
- DOI: https://doi.org/10.1090/S0025-5718-03-01541-2
- MathSciNet review: 2034110