Preconditioning spectral element schemes for definite and indefinite problems

Spectral element schemes for the solution of elliptic boundary value proble ms are considered. Preconditioning methods based on finite difference and f inite element schemes are implemented. Numerical experiments show that inve rting the preconditioner by a single multigrid iteration is most efficient and that the finite difference preconditioner is superior to the finite ele ment one for both definite and indefinite problems. A multigrid preconditio ner is also derived from the finite difference preconditioner and is found suitable for the CGS acceleration method. It is pointed out that, for the f inite difference and finite element preconditioners, CGS does not always co nverge to the accurate algebraic solution. (C) 1999 John Wiley & Sons, Inc.