A Legendre spectral Galerkin method for the biharmonic Dirichlet problem

A Legendre spectral Galerkin method is presented for the solution of the bi harmonic 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 approac h is used to reduce the resulting linear system to one involving the approx imation of the Laplacian of the solution on the two vertical sides of the s quare. 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 metho d.