On some theoretical and practical aspects of multigrid methods
Author:
R. A. Nicolaides
Journal:
Math. Comp. 33 (1979), 933952
MSC:
Primary 65N30
MathSciNet review:
528048
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Abstract 
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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.
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M. D. GUNZBURGER & R. A. NICOLAIDES. (In preparation.)
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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., SpringerVerlag, Berlin, 1977.
 [14]
A. JAMESON, Personal communication.
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A. Nicolaides, On multiple grid and related techniques for solving
discrete elliptic systems, J. Computational Phys. 19
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indefinite case, Math. Comp.
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T. CRAIG POLING, M.A. Thesis, College of William and Mary, Williamsburg, Virginia, 1977.
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V. Southwell, Relaxation Methods in Theoretical Physics,
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 [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. 89116. MR 0502025 (58:19215)
 [2]
 I. BABUŠKA, "The selfadaptive approach in the finite element method," Mathematics of Finite Elements and Applications (J. R. Whiteman, Ed.), Academic Press, London, pp. 125143.
 [3]
 I. BABUŠKA & W. RHEINBOLDT, Computational Aspects of Finite Element Analysis, Computer Science Technical Report TR518, University of Maryland, April, 1977, pp. 131.
 [4]
 I. BABUŠKA & W. RHEINBOLDT, Error Estimates for Adaptive Finite Element Computations, Inst. Phys. and Tech, Technical Note BN854, University of Maryland, May, 1977, pp. 141.
 [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. 861883. (Russian) MR 0215538 (35:6378)
 [6]
 A. BRANDT, MultiLevel 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, SpringerVerlag, Berlin, 1972, pp. 8289.
 [7]
 A. BRANDT, "Multilevel adaptive solution to boundary value problems," Math. Comp., v. 31, 1977, pp. 333391. MR 0431719 (55:4714)
 [8]
 A. BRANDT, MultiLevel Adaptive Techniques (MLAT): Ideas and Software, Proc. Conf. Mathematical Software; MRC, Wisconsin, 1977.
 [9]
 A. BRANDT & J. R. SOUTH, JR., Application of a MultiLevel Grid Method to Transonic Flow Calculations, ICASE Report No. 768, 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. 11031120. 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. 559564. (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., SpringerVerlag, 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. 418431. 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. 892906. MR 0488722 (58:8239)
 [17]
 R. A. NICOLAIDES, "On multigrid convergence in the indefinite case," Math. Comp., v. 32, 1978, pp. 10821086. 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)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197905280484
PII:
S 00255718(1979)05280484
Article copyright:
© Copyright 1979 American Mathematical Society
