Domain decomposition methods for eigenvalue problems. (English) Zbl 0952.65085
The author proposes several domain decomposition methods to compute the smallest eigenvalue of linear symmetric partial differential operators. The main idea is to determine an appropriate boundary condition at the interface separating two subdomains from the zero of a nonlinear operator. This concept is applied to overlapping and nonoverlapping schemes as well as to domain imbedding algorithms. Some numerical experiments for two-dimensional problems verify the convergence of the proposed algorithms.
Reviewer: Gunther Schmidt (Berlin)
MSC:
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 35P15 | Estimates of eigenvalues in context of PDEs |
Keywords:
domain decomposition; eigenvalue; fictitious domain; domain inbedding; Schwarz alternating method; numerical experiments; convergenceReferences:
| [1] | Arbenz, P.; Gander, W.; Golub, G. H., Restricted rank modification of the symmetric eigenvalue problem: theoretical considerations, Linear Algebra Appl., 104, 75-95 (1988) · Zbl 0652.15004 |
| [2] | Arbenz, P.; Golub, G. H., On the spectral decomposition of hermitian matrices modified by low rank perturbations with applications, SIAM J. Matrix. Anal. Appl., 9, 40-58 (1988) · Zbl 0646.65033 |
| [3] | Bennighof, J. K., Component mode iteration for frequency calculations, AIAA, 25, 7, 996-1002 (1987) · Zbl 0621.73059 |
| [4] | Borgers, C.; Widlund, O. B., On finite element domain imbedding methods, SIAM J. Numer. Anal., 27, 963-978 (1990) · Zbl 0705.65078 |
| [5] | Bourquin, F., Analysis and comparison of several component mode synthesis methods on one-dimensional domains, Numer. Math., 58, 11-34 (1990) · Zbl 0686.34026 |
| [6] | Bourquin, F., Component mode synthesis and eigenvalues of second order operators: discretization and algorithm, Math. Modelling Numer. Anal., 26, 385-423 (1992) · Zbl 0765.65100 |
| [7] | Bourquin, F.; d’Hennezel, F., Numerical study of an intrinsic component mode sysnthesis method, Comput. Methods Appl. Mech. Eng., 97, 49-76 (1992) · Zbl 0763.73080 |
| [8] | Buzbee, B. L.; Dorr, F. W.; George, J. A.; Golub, G. H., The direct solution of the discrete poisson equation on irregular regions, SIAM J. Numer. Anal., 8, 722-736 (1971) · Zbl 0231.65083 |
| [15] | Driscoll, T. A., Eigenmodes of isospectral drums, SIAM Rev., 39, 1-17 (1997) · Zbl 0949.65114 |
| [16] | Dryja, M., A finite element capacitance matrix method for elliptic problems on regions partitioned into substructures, Numer. Math., 44, 153-168 (1984) · Zbl 0568.65075 |
| [18] | D’Yakonov, E. G.; Knyazev, A. V., On an iterative method for finding lower eigenvalues, Russian J. Numer. Anal. Math. Modelling, 7, 6, 473-486 (1992) · Zbl 0816.65017 |
| [19] | Farhat, C.; Geradin, M., On a component mode synthesis method and its application to incompatible substructures, Comput. Struct, 51, 459-473 (1994) · Zbl 0900.73338 |
| [21] | Glowinski, R.; Pan, T.-W.; Periaux, J., A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations, Comput. Methods Appl. Mech. and Eng., 112, 133-148 (1994) · Zbl 0845.76069 |
| [23] | Knyazev, A. V., Convergence rate estimates for iterative methods for a mesh symmetric eigenvalue problem, Soviet J. Numer. Anal. Math. Modelling, 2, 371-396 (1987) · Zbl 0825.65034 |
| [24] | Knyazev, A. V.; Skorokhodov, A. L., Preconditioned iterative methods in subspace for solving linear systems with indefinite coefficient matrices and eigenvalue problems, Soviet J. Numer. Anal. Math. Modelling, 4, 4, 283-301 (1989) · Zbl 0825.65025 |
| [25] | Knyazev, A. V.; Skorokhodov, A. L., Preconditioned gradient-type iterative methods in a subspace for partial generalized symmetric eigenvalue problems, SIAM J. Numer. Anal., 31, 1225-1239 (1994) · Zbl 0812.65031 |
| [29] | Liew, K. M.; Hung, K. C.; Lim, M. K., On the use of the domain decomposition method for vibration of symmetric laminates having discontinuities at the same edge, J. Sound Vib., 178, 2, 243-264 (1994) |
| [31] | Lui, S. H., Kron’s method for symmetric eigenvalue problems, J. Comput. Appl. Math., 98, 35-48 (1998) · Zbl 0934.65046 |
| [34] | Marchuk, G. I.; Kuznetsov, Yu. A.; Matsokin, A. M., Fictitious domain and domain decomposition methods, Soviet J. Numer. Anal. Math. Modelling, 1, 3-35 (1986) · Zbl 0825.65027 |
| [38] | Simpson, A., Scanning Kron’s determinant, Quart. J. Mech. Appl. Math., 27, 27-43 (1974) · Zbl 0297.70014 |
| [41] | Le Tallec, P., Domain decomposition methods in computational mechanics, Comput. Mech. Adv., 1, 121-220 (1994) · Zbl 0802.73079 |
| [42] | Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Rev., 34, 581-613 (1992) · Zbl 0788.65037 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
