Higher order multigrid methods
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- Math. Comp. 43 (1984), 89-115 Request permission
Abstract:
This paper is concerned with the treatment of higher order multi-grid techniques for obtaining accurate finite difference approximations to partial differential equations. The three basic techniques considered are a multi-grid process involving smoothing via higher order difference approximations, iterated defect corrections with multi-grid used as an inner loop equation solver, and tau-extrapolation. Efficient versions of each of these three basic schemes are developed and analyzed by local mode analysis and numerical experiments. The numerical tests focus on fourth and sixth order discretizations of Poisson’s equations and demonstrate that the three methods performed similarly yet substantially better than the usual multi-grid method, even when the right-hand side lacked sufficient smoothness.References
-
W. Auzinger & H. J. Stetter, "Defect corrections and multigrid iterations." Preliminary report.
- J. H. Bramble and B. E. Hubbard, Approximation of derivatives by finite difference methods in elliptic boundary value problems, Contributions to Differential Equations 3 (1964), 399–410. MR 166935
- Achi Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977), no. 138, 333–390. MR 431719, DOI 10.1090/S0025-5718-1977-0431719-X
- Achi Brandt, Numerical stability and fast solutions to boundary value problems, Boundary and interior layers—computational and asymptotic methods (Proc. Conf., Trinity College, Dublin, 1980) Boole, Dún Laoghaire, 1980, pp. 29–49. MR 589349
- Achi Brandt and Nathan Dinar, Multigrid solutions to elliptic flow problems, Numerical methods for partial differential equations (Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 42, Academic Press, New York-London, 1979, pp. 53–147. MR 558216
- Lothar Collatz, The numerical treatment of differential equations. 3d ed, Die Grundlehren der mathematischen Wissenschaften, Band 60, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. Translated from a supplemented version of the 2d German edition by P. G. Williams. MR 0109436 H. Förster, K. Stüben & V. Trottenberg, "Non-standard multi-grid techniques using checkered relaxation and intermediate grid," in Elliptic Problem Solvers (M. Schultz, Ed.), Academic Press, New York, 1980.
- Wolfgang Hackbusch, Bemerkungen zur iterierten Defektkorrektur und zu ihrer Kombination mit Mehrgitterverfahren, Rev. Roumaine Math. Pures Appl. 26 (1981), no. 10, 1319–1329 (German). MR 646400
- Wolfgang Hackbusch, Survey of convergence proofs for multigrid iterations, Special topics of applied mathematics (Proc. Sem., Ges. Math. Datenverarb., Bonn, 1979) North-Holland, Amsterdam-New York, 1980, pp. 151–164. MR 585154 S. Schaffer, High Order Multi-Grid Methods to Solve the Poisson Equation, Proc. NASA-Ames Res. Center Symp. on Multigrid Methods, Moffett Field, Oct., 1981. S. Schaffer, Higher Order Multi-Grid Methods, Ph.D. Thesis, Colorado State University, May, 1982.
- Hans J. Stetter, The defect correction principle and discretization methods, Numer. Math. 29 (1977/78), no. 4, 425–443. MR 474803, DOI 10.1007/BF01432879 K. Stüben, Local Mode Analysis for the Solution of Elliptic Problems by Multigrid Methods, Internal Report, GMD-IMA, St. Augustin, Germany, 1982.
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Math. Comp. 43 (1984), 89-115
- MSC: Primary 65N05; Secondary 65N10, 65N50
- DOI: https://doi.org/10.1090/S0025-5718-1984-0744926-7
- MathSciNet review: 744926