A Legendre Spectral Collocation Method for the Biharmonic Dirichlet Problem
Department of Mathematical and Computer Sciences,
Colorado School of Mines, Golden, Colorado 80401, U.S.A. (firstname.lastname@example.org)
2 Department of Mathematics and Statistics, University of Cyprus, P.O. Box 537, 1678 Nicosia, Cyprus.
Revised: 23 November 1999
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.
Mathematics Subject Classification: 65N35 / 65N22
Key words: Biharmonic Dirichlet problem / spectral collocation / Schur complement / preconditioned conjugate gradient method.
© EDP Sciences, SMAI, 2000