## Preconditioned minimal residual methods for Chebyshev spectral calculations.(English)Zbl 0615.65118

Preconditioned iterative methods are considered for the linear systems of algebraic equations which arise when elliptic problems of second order are discretized by spectral methods. While these methods often are very accurate, they lead to dense, rather illconditioned coefficient matrices. The techniques considered here are based on preconditioning the operator with low accuracy finite difference and finite element approximations or by incomplete Cholesky factorizations of the corresponding sparse matrices. It should be noted the fast Fourier transforms can be used to find the matrix vector products required to compute the residuals related to the original models.
In this quite carefully prepared paper, a number of iterative methods are studied. Of these is a normal equation version of the conjugate gradient method. The best results are obtained by version of a stationary second- degree method called the DuFort-Frankel method by the authors. A minimal residual strategy is used in which the two parameters are determined dynamically. A number of numerical results are given.
Reviewer: O.Widlund

### MSC:

 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 35J25 Boundary value problems for second-order elliptic equations 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65F35 Numerical computation of matrix norms, conditioning, scaling
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