A legendre spectral collocation method for the biharmonic Dirichlet problem

A Legendre spectral collocation method is presented for the solution of the biharmonic Dirichlet problem on a square. The solution and its Laplacian a re approximated using the set of basis functions suggested by Shen, which a re linear combinations of Legendre polynomials. A Schur complement approach is used to reduce the resulting linear system to one involving the approxi mation of the Laplacian of the solution on the two vertical sides of the sq uare. The Schur complement system is solved by a preconditioned conjugate g radient method. The total cost of the algorithm is O(N-3). Numerical result s demonstrate the spectral convergence of the method. Mathematics Subject C lassification. 65N35, 65N22.