A Legendre Spectral Galerkin Method for the Biharmonic Dirichlet Problem SIAM Journal on Scientific Computing Volume 22, Number 5 pp. 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.
Numerical results demonstrate the spectral convergence of the method.
65N35, 65N22 PII .