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Parallel Poisson and Biharmonic solvers

Parallele Methoden zur Lösung von Poisson und Biharmonischen Gleichungen

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Abstract

In this paper we develop direct and iterative algorithms for the solution of finite difference approximations of the Poisson and Biharmonic equations on a square, using a number of arithmetic units in parallel. Assuming ann×n grid of mesh points, we show that direct algorithms for the Poisson and Biharmonic equations require 0(logn) and 0(n) time steps, respectively. The corresponding speedup over the sequential algorithms are 0(n 2) and 0(n 2logn). We also compare the efficiency of these direct algorithms with parallel SOR and ADI algorithms for the Poisson equation, and a parallel semi-direct method for the Biharmonic equation treated as a coupled pair of Poisson equations.

Zusammenfassung

In diesem Artikel entwickeln wir direkte und iterative Algorithmen für die Lösung von Differenzen-Approximationen der Poisson und Biharmonischen Gleichungen über einem Quadrat, wobei eine Anzahl paralleler Arithmetikeinheiten verwendet wird. Unter der Annahme eines [n, n] Netzes zeigen wir, daß direkte Algorithmen für die Poisson bzw. Biharmonischen Gleichungen 0(logn) bzw. 0(n) Schritte benötigen. Der entsprechende Gewinn über die sequentiellen Algorithmen beträgt 0(n 2) bzw. 0(n 2logn). Wir vergleichen außerdem die Effizienz dieser direkten Algorithmen mit parallelen SOR und ADI Algorithmen für die Poisson Gleichung, und mit einer parallelen halb-direkten Methode für die Behandlung der Biharmonischen Gleichung als eines gekoppelten Paares von Poisson Gleichungen.

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Authors and Affiliations

  1. Center for Advanced Computation, University of Illinois, 61801, Urbana, IL, USA

    A. H. Sameh

  2. Department of Computer Science, University of Illinois, 61801, Urbana, IL, USA

    A. H. Sameh,  S. C. Chen &  D. J. Kuck

Authors
  1. A. H. Sameh
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  2. S. C. Chen
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  3. D. J. Kuck
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Additional information

This work was supported in part by NSF Grant DCR 73-07980 A02; and in part by the Advanced Research Project Agency of the Department of Defense under Contract No. DAHC04-72-C-0001.

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Sameh, A.H., Chen, S.C. & Kuck, D.J. Parallel Poisson and Biharmonic solvers. Computing 17, 219–230 (1976). https://doi.org/10.1007/BF02259647

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  • DOI: https://doi.org/10.1007/BF02259647

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