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Improved spectral multigrid methods for periodic elliptic problems. (English) Zbl 0569.65084

The authors introduce for the periodic case some refinements which can substantially improve the overall efficiency of the spectral multigrid method. It is shown that by evaluating the fluxes at the midpoints between the usual collocation points a ”midpoint” pseudospectral discretization is obtained which eliminates the need for filtering the highest modes, resulting in improved accuracy and efficiency. For the isotropic case a weighted residual relaxation scheme with a greatly improved smoothing rate is introduced and it is shown that by property scaling the relaxation parameters, the smoothing rates obtained for constant coefficient problems also hold even when the coefficients are not constant. Some comparisons of choices for coarse grid operators and residual transfers are also made. The authors introduce another relaxation scheme, based on defect corrections, which is efficient even for highly anisotropic problems. Numerical results are given.
Reviewer: K.Najzar

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35B10 Periodic solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
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References:

[1] Gottlieb, D.; Orszag, S. A., Numerical Analysis of Spectral Methods, (NSF-CBMS Monograph No. 26 (1977), Soc. Ind. and Appl. Math: Soc. Ind. and Appl. Math Philadelphia) · Zbl 0561.76076
[2] Orszag, S. A., J. Comput. Phys., 37, 70 (1980) · Zbl 0476.65078
[3] Wong, Y. S.; Zang, T. A.; Hussaini, M. Y., Efficient Iterative Techniques for the Solution of Spectral Equations, (Symposium on Spectral Methods in Partial Differential Equations (August 16-18, 1982), ICASE, NASA Langley Research Center: ICASE, NASA Langley Research Center Hampton, Va)
[4] Zang, T. A.; Wong, Y. S.; Hussaini, M. Y., J. Comput. Phys., 48, 485 (1982) · Zbl 0496.65061
[5] T. A. Zang, Y. S. Wong, AND M. Y. HussainiJ. Comput. Phys.; T. A. Zang, Y. S. Wong, AND M. Y. HussainiJ. Comput. Phys. · Zbl 0543.65071
[6] Brandt, A., Math. Comput., 31, 333 (1977) · Zbl 0373.65054
[7] Brandt, A., (Hackbusch, W.; Trottenberg, U., Multigrid Methods (1982), Springer-Verlag: Springer-Verlag New York)
[8] Kaufman, E. H.; Taylor, G. D., Internat. J. Numer. Methods Engrg., 9, 297 (1975) · Zbl 0304.65014
[9] McCrory, R. L.; Orszag, S. A., J. Comput. Phys., 37, 93 (1980) · Zbl 0451.60076
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