×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Axelsson, O., On preconditioning and convergence acceleration in sparse matrix problems, CERN report 74-10, (1974), Geneva, Switzerland · Zbl 0354.65020
[2] Canuto, C.; Quarteroni, A., (), 55-78
[3] Concus, P.; Golub, G.G., (), 56-65
[4] Dupont, T.; Kendall, R.P.; Rachford, H.H., SIAM J. numer. anal., 5, 559-573, (1968)
[5] {\scD. Funaro}, RAIRO Anal. Numer., in press.
[6] Glowinski, R.; Mantel, B.; Periaux, J.; Pironneau, O.; Poirier, G., (), 445-487
[7] Gottlieb, D.; Gustafsson, B., SIAM J. numer. anal., 13, 129-144, (1976)
[8] Gottlieb, D.; Lustman, L., Comput. fluids, 11, No. 2, 107-120, (1983)
[9] Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods: theory and applications, () · Zbl 0561.76076
[10] Haidvogel, D.B.; Zang, T.A., J. comput. phys., 30, 167-180, (1979)
[11] Meijerink, J.A.; van der Vorst, H.A., Math. comp., 31, 148-162, (1977)
[12] Meijerink, J.A.; van der Vorst, H.A., J. comput. phys., 44, 134-155, (1981)
[13] Orszag, S.A., J. comput. phys., 37, 70-92, (1980)
[14] Peyret, R.; Taylor, T.D., Computational methods for fluid flow, (1982), Springer New York
[15] Vinsome, P.K.W., (), 149-159
[16] ()
[17] Young, D.M., J. approx. theory, 5, 137-149, (1971)
[18] Young, D.M., Iterative solutions of large linear systems, (1971), Academic Press New York
[19] Young, D.M.; Jea, K.C., Linear algebra appl., 34, 159-194, (1980)
[20] Widlund, O., SIAM J. numer. anal., 15, 801-812, (1978)
[21] Zang, T.A.; Wong, Y.S.; Hussaini, M.Y., J. comput. phys., 48, 485-501, (1982)
[22] Haldenwang, P.; Labrosse, G.; Abboudi, S.; Deville, M., J. comput. phys., 55, 115-125, (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.