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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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