×

Multigrid methods for combined finite difference and Fourier problems. (English) Zbl 0657.65118

Considered are combinations of finite difference and pseudo-spectral approximations for 2D and 3D elliptic problems with periodic and first or second kind boundary conditions. This is of interest since the usual combination of Fourier with Chebyshev approximation not only has a condition number greater by two orders but also may result (for not appropriately chosen smoothing iterations) into loss of convergence.
The paper contains a description of several details of multigrid methods to solve the discretized equations, lists parameters (being optimal in case of constant coefficients) for the weighted residual relaxation of A. Brandt [Lect. Notes Math. 960, 220-312 (1982; Zbl 0505.65037)], and shows much numerical results.
Reviewer: G.Stoyan

MSC:

65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
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

Citations:

Zbl 0505.65037
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Brandt, A., Math. comput., 31, 333, (1977)
[2] Brandt, A., ()
[3] Brandt, A.; Fulton, S.R.; Taylor, G.D., J. comput, phys., 58, 96, (1985)
[4] Canuto, C.; Quarteroni, A., Math. comput., 38, 67, (1982)
[5] Canuto, C.; Maday, Y.; Quarteroni, A., Numer. math., 39, 205, (1982)
[6] Canuto, C.; Quarteroni, A., (), 55
[7] Canuto, C., SIAM J. numer. anal., 23, 815, (1986)
[8] Collatz, L., The numerical treatment of differential equations, (1966), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0221.65088
[9] De Boor, C., A practical guide to splines, (1978), Springer-Verlag Berlin/Heidelberg/New York · Zbl 0406.41003
[10] Erlebacher, G.; Zang, T.A.; Hussaini, M.Y., (), (unpublished)
[11] Faddejew, D.K.; Faddejewa, W.M., Numerische methoden der linearen algebra, (1979), Oldenbourg Verlag Munich/Vienna · Zbl 0119.12202
[12] Hackbusch, W., Theorie and numerik elliptischer differentialgleichungen, (1986), Teubner Studienbücher Stuttgart · Zbl 0609.65065
[13] {\scW. Heinrichs} Line relaxation for spectral multigrid methods, J. Comput. Phys., in press.
[14] Kaufman, E.H.; Taylor, G.D., Int. J. num. methods engrg., 9, 297, (1975)
[15] Orszag, S.A., J. comput. phys., 37, 70, (1980)
[16] Streett, C.L.; Zang, T.A.; Hussaini, M.Y., J. comput. phys., 57, 42, (1985)
[17] Stüben, K.; Trottenberg, U., ()
[18] Zang, T.A.; Wong, Y.S.; Hussaini, M.Y., J. comput. phys., 48, 485, (1982)
[19] Zang, T.A.; Wong, Y.S.; Hussaini, M.Y., J. comput. phys., 54, 499, (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.