A Legendre Spectral Galerkin Method for the Biharmonic Dirichlet Problem
Authors:
- Bernard BialeckiBBBernard Bialecki
- Andreas KarageorghisAKAndreas Karageorghis
- 1 January 2001
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Scientific Computing
- Vol. 22 (5) , 1549-1569
Abstract
A Legendre spectral Galerkin 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 two 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 or the Cholesky method. The total cost of the algorithm is O(N 3 ). Numerical results demonstrate the spectral convergence of the method.Keywords
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