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


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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[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., (Voigt, Robert G.; Gottlieb, David; Yousuff Hussaini, M., Spectral Methods for Partial Differential Equations (1984), SIAM: SIAM Philadelphia), 55-78
[3] Concus, P.; Golub, G. G., (Glowinski, R.; Lions, J. L., Lecture Notes in Economics and Mathematical Systems, Vol. 134 (1976), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), 56-65
[4] Dupont, T.; Kendall, R. P.; Rachford, H. H., SIAM J. Numer. Anal., 5, 559-573 (1968) · Zbl 0174.47603
[6] Glowinski, R.; Mantel, B.; Periaux, J.; Pironneau, O.; Poirier, G., (Glowinski, R.; Lions, J. L., Computing Methods in Applied Sciences and Engineering (1980), NorthHolland: NorthHolland Amsterdam/New York), 445-487 · Zbl 0445.65098
[7] Gottlieb, D.; Gustafsson, B., SIAM J. Numer. Anal., 13, 129-144 (1976) · Zbl 0344.35048
[8] Gottlieb, D.; Lustman, L., Comput. Fluids, 11, No. 2, 107-120 (1983) · Zbl 0511.76003
[9] Gottlieb, D.; Orszag, S. A., Numerical Analysis of Spectral Methods: Theory and Applications, (CBMS Regional Conference Series in Applied Mathematics (1977), SIAM: SIAM Philadelphia) · Zbl 0561.76076
[10] Haidvogel, D. B.; Zang, T. A., J. Comput. Phys., 30, 167-180 (1979) · Zbl 0397.65077
[11] Meijerink, J. A.; van der Vorst, H. A., Math. Comp., 31, 148-162 (1977) · Zbl 0349.65020
[12] Meijerink, J. A.; van der Vorst, H. A., J. Comput. Phys., 44, 134-155 (1981) · Zbl 0472.65028
[13] Orszag, S. A., J. Comput. Phys., 37, 70-92 (1980) · Zbl 0476.65078
[14] Peyret, R.; Taylor, T. D., Computational Methods for Fluid Flow (1982), Springer: Springer New York
[15] Vinsome, P. K.W., (Proc. 4th Sympos. Reservoir Simulation (1976), Society of Petroleum Engineers of AIME: Society of Petroleum Engineers of AIME New York), 149-159
[16] (Voigt, R. G.; Gottlieb, D.; Hussaini, M. Y., Spectral Methods for Partial Differential Equations (1984), SIAM: SIAM Philadelphia)
[17] Young, D. M., J. Approx. Theory, 5, 137-149 (1971)
[18] Young, D. M., Iterative Solutions of Large Linear Systems (1971), Academic Press: Academic Press New York · Zbl 0231.65034
[19] Young, D. M.; Jea, K. C., Linear Algebra Appl., 34, 159-194 (1980) · Zbl 0463.65025
[20] Widlund, O., SIAM J. Numer. Anal., 15, 801-812 (1978) · Zbl 0398.65030
[21] Zang, T. A.; Wong, Y. S.; Hussaini, M. Y., J. Comput. Phys., 48, 485-501 (1982) · Zbl 0496.65061
[22] Haldenwang, P.; Labrosse, G.; Abboudi, S.; Deville, M., J. Comput. Phys., 55, 115-125 (1984) · Zbl 0544.65071
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