Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Higher order multigrid methods


Author: Steve Schaffer
Journal: Math. Comp. 43 (1984), 89-115, S1
MSC: Primary 65N05; Secondary 65N10, 65N50
DOI: https://doi.org/10.1090/S0025-5718-1984-0744926-7
MathSciNet review: 744926
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] W. Auzinger & H. J. Stetter, "Defect corrections and multigrid iterations." Preliminary report.
  • [2] J. H. Bramble & B. E. Hubbard, "Approximation of derivatives by finite difference methods in elliptic boundary value problems," Contrib. Differential Equations, v. 3, 1963, pp. 399-410. MR 0166935 (29:4208)
  • [3] A. Brandt, "Multi-level adaptive solutions to boundary value problems," Math. Comp., v. 31, 1977, pp. 333-390. MR 0431719 (55:4714)
  • [4] A. Brandt, "Numerical stability and fast solutions to boundary value problems," in Boundary and Interior Layers--Computational and Asymptotic Methods (J. J. H. Miller, Ed.), Boole Press, Dublin, 1980. MR 589349 (83b:65100)
  • [5] A. Brandt & N. Dinar, "Multi-grid solutions to elliptic flow problems," in Numerical Methods for Partial Differential Equations (V. S. Parter, Ed.), Academic Press, New York, 1979. MR 558216 (81a:65094)
  • [6] L. Collatz, Numerical Treatment of Differential Equations, 3rd ed., Springer-Verlag, Berlin, 1960. MR 0109436 (22:322)
  • [7] 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.
  • [8] W. Hackbusch, "Bemerkungen zur iterierten Defektkorrektur und zu ihrer Kombination mit Mehrgitterverfahren," Rev. Roumaine Math. Pures Appl., v. 26, 1981, pp. 1319-1329. MR 646400 (83c:65255)
  • [9] W. Hackbusch, "Survey of convergence proofs for multi-grid iterations," in Special Topics of Applied Mathematics (J. Frehse, et. al., Eds.), North-Holland, Amsterdam, 1980. MR 585154 (82j:65072)
  • [10] 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.
  • [11] S. Schaffer, Higher Order Multi-Grid Methods, Ph.D. Thesis, Colorado State University, May, 1982.
  • [12] H. J. Stetter, "The defect correction principle and discretization methods," Numer. Math., v. 29, 1978, pp. 425-443. MR 0474803 (57:14436)
  • [13] K. Stüben, Local Mode Analysis for the Solution of Elliptic Problems by Multigrid Methods, Internal Report, GMD-IMA, St. Augustin, Germany, 1982.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N05, 65N10, 65N50

Retrieve articles in all journals with MSC: 65N05, 65N10, 65N50


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1984-0744926-7
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society