A legendre spectral collocation method for the biharmonic Dirichlet problem
Date
2000ISSN
0764-583XSource
Mathematical Modelling and Numerical AnalysisVolume
34Issue
3Pages
637-662Google Scholar check
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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.