Abstract
This paper presents a fast iterative algorithm for the solution of a finite difference approximation of the biharmonic boundary value problem on a rectangular region. For solving this problem, the matrix decomposition algorithm is efficiently applied to the semi-direct method which essentially treats the biharmonic equation as a coupled system of Poisson equations. Assuming anN×N grid of mesh points, the number of operations required for one iteration and for the solution terminated by 0 (N −2) is 0 (N 2) and 0 (N 5/2 log2 N), respectively. ForN 2 processors, the parallel version of this algorithm would require 14 log2 N steps per iteration. Both results are better than those known. A numerical experiment in a serial computation is also given.
Zusammenfassung
In diesem Artikel wird ein schneller Algorithmus für die Lösung von Differenzen-Approximation des ersten Biharmonischen Randwertproblems auf einem rechtwinkligen Gebiet dargestellt. Für die Lösung dieses Problems wird der Algorithmus der Matrix-Dekomposition in der halbdirekten Methode, die die Biharmonische Gleichung als ein gekoppltes Paar von Poisson-Gleichungen behandelt, effektiv verwendet. Unter der Annahme einesNxN Netzes ist die Anzahl der erforderlichen arithmetischen Operationen für eine Iteration bzw. für eine Lösung, die mit Genauigkeit 0 (N −2) bestimmt wird, 0 (N 2) bzw. 0 (N 5/2 log2 N). Die parallele Version dieses Algorithmus fordert mitN 2 Prozessoren 14 log2 N Schritten für eine Iteration. Beide Resultate sind besser als bis jetzt bekannte Resultate. Für die sequentielle Berechnung wird ein numerisches Experiment angegeben.
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Vajteršic, M. A fast algorithm for solving the first biharmonic boundary value problem. Computing 23, 171–178 (1979). https://doi.org/10.1007/BF02252095
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DOI: https://doi.org/10.1007/BF02252095