On some theoretical and practical aspects of multigrid methods
Author:
R. A. Nicolaides
Journal:
Math. Comp. 33 (1979), 933-952
MSC:
Primary 65N30
DOI:
https://doi.org/10.1090/S0025-5718-1979-0528048-4
MathSciNet review:
528048
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: A description and explanation of a simple multigrid algorithm for solving finite element systems is given. Numerical results from an implementation are reported for a number of elliptic equations, including cases with singular coefficients and indefinite equations. The method shows the high efficiency, essentially independent of the grid spacing, predicted by the theory.
- [1] I. BABUŠKA, "Homogenization and its application," Mathematical and Computational Problems, Numerical Solution of Partial Differential Equations, III (SYNSPADE 1975), Academic Press, New York, 1976, pp. 89-116. MR 0502025 (58:19215)
- [2] I. BABUŠKA, "The self-adaptive approach in the finite element method," Mathematics of Finite Elements and Applications (J. R. Whiteman, Ed.), Academic Press, London, pp. 125-143.
- [3] I. BABUŠKA & W. RHEINBOLDT, Computational Aspects of Finite Element Analysis, Computer Science Technical Report TR-518, University of Maryland, April, 1977, pp. 1-31.
- [4] I. BABUŠKA & W. RHEINBOLDT, Error Estimates for Adaptive Finite Element Computations, Inst. Phys. and Tech, Technical Note BN-854, University of Maryland, May, 1977, pp. 1-41.
- [5] N. S. BAKHVALOV, "On the convergence of a relaxation method under natural constraints on an elliptic operator," Ž. Vyčisl. Mat. i Mat. Fiz., v. 6, 1966, pp. 861-883. (Russian) MR 0215538 (35:6378)
- [6] A. BRANDT, Multi-Level Adaptive Technique (MLAT) for Fast Numerical Solution to Boundary Value Problems, Proc. 3rd Internat. Conf. on Numerical Methods Fluid Mechanics (Paris, 1972); Lecture Notes in Physics, Vol. 18, Springer-Verlag, Berlin, 1972, pp. 82-89.
- [7] A. BRANDT, "Multi-level adaptive solution to boundary value problems," Math. Comp., v. 31, 1977, pp. 333-391. MR 0431719 (55:4714)
- [8] A. BRANDT, Multi-Level Adaptive Techniques (MLAT): Ideas and Software, Proc. Conf. Mathematical Software; MRC, Wisconsin, 1977.
- [9] A. BRANDT & J. R. SOUTH, JR., Application of a Multi-Level Grid Method to Transonic Flow Calculations, ICASE Report No. 76-8, 1976.
- [10] P. CONCUS & G. GOLUB, "Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 1103-1120. MR 0341890 (49:6636)
- [11] R. P. FEDERENKO, "The speed of convergence of an iteration process," Ž. Vyčisl. Mat. i Mat. Fiz., v. 4, 1964, pp. 559-564. (Russian) MR 0182163 (31:6386)
- [12] M. D. GUNZBURGER & R. A. NICOLAIDES. (In preparation.)
- [13] W. HACKBUSCH, "A fast iterative method for solving Poisson's equation in a general region," Numerical Treatment of Differential Equations (R. Bulirsch et al., Eds.), Lecture Notes in Math., Springer-Verlag, Berlin, 1977.
- [14] A. JAMESON, Personal communication.
- [15] R. A. NICOLAIDES, "On multiple grid and related techniques for solving discrete elliptic systems," J. Computational Phys., v. 19, 1975, pp. 418-431. MR 0413541 (54:1655)
- [16]
R. A. NICOLAIDES, "On the
convergence of an algorithm for solving finite element equations," Math. Comp., v. 31, 1977, pp. 892-906. MR 0488722 (58:8239)
- [17] R. A. NICOLAIDES, "On multigrid convergence in the indefinite case," Math. Comp., v. 32, 1978, pp. 1082-1086. MR 0520340 (58:25009)
- [18] T. CRAIG POLING, M.A. Thesis, College of William and Mary, Williamsburg, Virginia, 1977.
- [19] R. V. SOUTHWELL, Relaxation Methods in Theoretical Physics, Clarendon Press, Oxford, 1946. MR 0018983 (8:355f)
Retrieve articles in Mathematics of Computation with MSC: 65N30
Retrieve articles in all journals with MSC: 65N30
Additional Information
DOI:
https://doi.org/10.1090/S0025-5718-1979-0528048-4
Article copyright:
© Copyright 1979
American Mathematical Society