Preconditioning Chebyshev spectral collocation method for elliptic partial differential equations. (English) Zbl 0861.65095

Kim, Sang Dong; Parter, Seymour V.

SIAM J. Numer. Anal. 33, No. 6, 2375-2400 (1996).

A preconditioning technique for the solution of Chebyshev spectral collocation equations with Dirichlet boundary conditions is analyzed. Bounds for the eigenvalues of the Helmholtz equation are derived. A bilinear finite element preconditioner is proposed where the stiffness matrix is associated with the Chebyshev weight. Hence the preconditioner works for the weighted collocation matrix. Bounds on the condition number are derived in the weighted Sobolev space \(H^1_{0,w}\) for general elliptic operators.

Reviewer: W.Heinrichs (Essen)

MSC:

65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65F35	Numerical computation of matrix norms, conditioning, scaling

Keywords:

Chebyshev-Gauss-Lobatto method; preconditioning; Chebyshev spectral collocation; Helmholtz equation; finite element; condition number