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