Abstract
We present a new approach to the construction of Domain Decomposition (DD) preconditioners for the conjugate gradient method applied to the solution of symmetric and positive definite finite element equations. The DD technique is based on a non-overlapping decomposition of the domain Ω intop subdomains connected later with thep processors of a MIMD computer. The DD preconditioner derived contains three block matrices which must be specified for the specific problem considered. One of the matrices is used for the transformation of the nodal finite element basis into the approximate discrete harmonic basis. The other two matrices are block preconditioners for the Dirichlet problems arising on the subdomains and for a modified Schur complement defined over all nodes on the coupling boundaries between the subdomains. The relative spectral condition number is estimated. Relations to the additive Schwarz method are discussed. In the second part of this paper, we will apply the results of this paper to two-dimensional, symmetric, second-order, elliptic boundary value problems and present numerical results performed on a transputer-network.
Zusammenfassung
In der vorliegenden Arbeit wird ein neuer Zugang zur Konstruktion von Vorkonditionierungsoperatoren auf der Basis von Gebietsdekompositionstechniken (DD Techniken) beschrieben. Anwendungen finden diese DD Vorkonditionierungen im Verfahren der konjugierten Gradienten zur iterativen Lösung von symmetrischen und positiv definiten Finiten-Elemente Gleichungen. Die DD Technik basiert auf einer Zerlegung des Gebietes Ω inp sich nicht überlappende Teilgebiete, die später denp Prozessoren eines MIMD Rechners zugeordnet sind. Die DD Vorkonditionierung enthält drei Blockmatrizen, die für ein konkretes Anwendungsproblem jeweils zu spezifizieren sind. Eine dieser Matrizen wird genutzt, um die Knotenbasis in eine näherungsweise diskret harmonische Basis zu transformieren. Die anderen beiden Matrizen können als Blockvorkonditionierungen für die in jedem Teilgebiet entstehenden Dirichlet-Probleme und für ein modifiziertes Schurkomplement auf den Knoten der Koppelränder zwischen den Teilgebieten interpretiert werden. Die relative spektrale Konditionszahl wird abgeschätzt. Eine direkte Verbindung der vorgeschlagenen DD Vorkonditionierung zu einer Additiven Schwarzschen Methode kann gezeigt werden. Im zweiten Teil dieser Artikelserie werden die Resultate dieser Arbeit auf ebene, symmetrische Randwertprobleme für partielle Differentialgleichungen zweiter Ordnung angewandt und die numerischen Resultate, die auf einem Transputer-Hypercube erzeugt wurden, diskutiert.
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Haase, G., Langer, U. & Meyer, A. The approximate Dirichlet Domain Decomposition method. Part I: An algebraic approach. Computing 47, 137–151 (1991). https://doi.org/10.1007/BF02253431
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DOI: https://doi.org/10.1007/BF02253431