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A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem

Published online by Cambridge University Press:  15 April 2002

Bernard Bialecki  and
Andreas Karageorghis
Bernard Bialecki
Affiliation:
Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401, U.S.A. (bbialeck@mines.edu)
Andreas Karageorghis
Affiliation:
Department of Mathematics and Statistics, University of Cyprus, P.O. Box 537, 1678 Nicosia, Cyprus.
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Abstract

A Legendre spectral collocation method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian are approximated using the set of basis functions suggested by Shen, which are linear combinations of Legendre polynomials. A Schur complement approach is used to reduce the resulting linear system to one involving the approximation of the Laplacian of the solution on the two vertical sides of the square. The Schur complement system is solved by a preconditioned conjugate gradient method. The total cost of the algorithm is O(N3). Numerical results demonstrate the spectral convergence of the method.

Keywords

Biharmonic Dirichlet problemspectral collocationSchur complementpreconditioned conjugate gradient method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 3 , May 2000 , pp. 637 - 662
Copyright
© EDP Sciences, SMAI, 2000

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