Abstract
In the first part of this article series, we had derived Domain Decomposition (DD) preconditioners containing three block matrices which must be specified for specific applications. In the present paper, we consider finite element equations arising from the DD discretization of plane, symmetric, 2nd-order, elliptic b.v.p.s and specify the matrices involved in the preconditioner via multigrid and hierarchical techniques. The resulting DD-PCCG methods are asymptotically almost optimal with respect to the operation count and well suited for parallel computations on MIMD computers with local memory and message passing. The numerical experiments performed on a transputer hypercube confirm the efficiency of the DD preconditioners proposed.
Zusammenfassung
Im ersten Teil dieser Artikelserie haben wir auf Basis von Gebietsdekompositionstechniken (DD Techniken) Vorkonditionierungsoperatoren konstruiert. Diese DD Vorkonditionierungen enthalten drei Blockmatrizen, die für spezifische Anwendungsfälle zu konkretisieren sind. In der vorliegenden Arbeit betrachten wir Finite-Elemente-Gleichungen, die bei der DD Diskretisierung von ebenen, symmetrischen, elliptischen Randwertproblemen für partielle Differentialgleichungen zweiter Ordnung entstehen. Zur Definition der oben genannten Blockmatrizen werden Mehrgitter-und hierarchische Techniken herangezogen. Die entstehenden DD-PCCCG Verfahren sind bezüglich des arithmetischen Aufwands asymptotisch fast optimal und bestens zur Parallelrechnung auf MIMD-Computern mit lokalem Speicher und Botschaftenaustausch geeignet. Die auf einem Transputer-Hypercube durchgeführten numerischen Experimente belegen nachhaltig die Effektivität der vorgeschlagenen DD Vorkonditionierungen.
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References
Bjørstad, P. E., Widlund, O. B.: Iterative methods for the solution of elliptic problems on region partitioned into substructures. SIAM J. Numer. Anal.,23, 1097–1120, (1986).
Börgers, C.: The Neumann-Dirichlet domain decomposition method with inexact solvers on the subdomains. Numer. Math.,55, 123–136 (1989).
Bramble, J. H., Pasciak, J. E., Schatz, A. H.: The construction of preconditioners for elliptic problems by substructuring I. Math. Comput. I:47, 103–134 (1986).
Chan, T. F.: Analysis of preconditioners for domain decomposition. SIAM J. Numer. Anal.,24, 382–390 (1987).
Chan, T. F., Glowinski, R., Periaux, J., Widlund, O. B. (eds.): Domain decomposition methods for PDEs. Proceedings of the 2nd Internationel Symposium on Domain Decomposition Methods, SIAM, Philadelphia 1989.
Chan, T. F., Glowinski, R., Periaux, J., Widlund, O. B. (eds.): Domain decomposition methods for PDEs. Proceedings of the 3rd Internationel Symposium on Domain Decomposition Methods, SIAM, Philadelphia 1990.
Chan, T. F., Resasco, D. C.: A framework for the analysis and construction of domain decomposition preconditioners. In [11] Domain decomposition methods for PDEs. Proceedings of the 1st International Symposium on Domain Decomposition Methods, SIAM, Philadelphia 1988.
Dryja, M.: A capacitance matrix method for Dirichlet problems on polygon regions. Numer. Math.,39, 51–64 (1982).
Dryja, M.: A finite element-capacitance method for elliptic problems on regions partioned into subregions. Numer. Math.,44, 153–165 (1984).
Globisch, G., Langer, U.: On the use of multigrid preconditioners in a multigrid software package. Proceedings of the 4th GDR Multigrid Seminar held at Unterwirbach, May 2–5, 1989, Report R-MATH-03/90 of the Institute of Mathematics of the Academy of Science, pp.
Glowinski, R., Golub, G. H., Meurant, G. A., Périaux, J.: (eds.): Domain decomposition methods for PDEs. Proceedings of the 1st International Symposium on Domain Decomposition Methods, SIAM, Philadelphia 1988.
Golub, G. H., Mayers, D.: The use of preconditioning over irregular regions. Lecture at 6th Int. Conf. on Computing Methods in Applied Sciences and Engineering, Versailles, 1983.
Haase, G., Langer, U., Meyer A.: Domain decomposition preconditioners with inexact subdomain solvers (appears in Journal of Numerical Linear Algebra with Applications).
Haase, G., Langer, U., Meyer, A.: A new approach to the Dirichlet domain decomposition method. Technical Report (in: Report Series R-MATH-09/90 of the Institute of Mathematics of the Academy of Science, Berlin, 59 pp.
Haase, G., Langer U., Meyer, A.: The approximate Dirichlet domain decomposition method. Part I: An algebraic approach. Computing47, 137–152.
Hackbusch, W.: Multigrid methods and applications. Springer Series in Computational Mathematics 4, Berlin, Heidelberg, New York, Tokio: Springer 1985.
Jung, M., Langer U., Meyer, A., Queck, W.,Schneider, M.: Multigrid preconditioners and their applications. Proceedings of the 3rd GDR Multigrid Seminar held at Berlin/Biesenthal, May 2–6, 1988, Report R-MATH-03/89 of the Institute of Mathematiks of the Academy of Science, pp. 11–52, Berlin 1989.
Przemieniecki, J. S.: Matrix structural analysis of substructures. AIAA J.,1, 138–147 (1963).
Schwarz, H. A.: Über einige Abbildungsaufgaben. Ges. Math. Abh.,11, 65–83 (1869).
Yserentant, H.: On the multi-level splitting of finite element spaces. Numer. Math.,49, 379–412 (1986).
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Haase, G., Langer, U. & Meyer, A. The approximate Dirichlet Domain Decomposition method. Part II: Applications to 2nd-order Elliptic B.V.P.s. Computing 47, 153–167 (1991). https://doi.org/10.1007/BF02253432
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DOI: https://doi.org/10.1007/BF02253432