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]**Ivo Babuška,*Homogenization and its application. Mathematical and computational problems*, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 89–116. MR**0502025****[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. Bahvalov,*Convergence of a relaxation method under natural constraints on an elliptic operator*, Ž. Vyčisl. Mat. i Mat. Fiz.**6**(1966), 861–883 (Russian). MR**0215538****[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]**Achi Brandt,*Multi-level adaptive solutions to boundary-value problems*, Math. Comp.**31**(1977), no. 138, 333–390. MR**0431719**, https://doi.org/10.1090/S0025-5718-1977-0431719-X**[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]**Paul Concus and Gene H. Golub,*Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations*, SIAM J. Numer. Anal.**10**(1973), 1103–1120. MR**0341890**, https://doi.org/10.1137/0710092**[11]**R. P. Fedorenko,*On the speed of convergence of an iteration process*, Ž. Vyčisl. Mat. i Mat. Fiz.**4**(1964), 559–564 (Russian). MR**0182163****[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.**19**(1975), no. 4, 418–431. MR**0413541****[16]**R. A. Nicolaides,*On the 𝑙² convergence of an algorithm for solving finite element equations*, Math. Comp.**31**(1977), no. 140, 892–906. MR**0488722**, https://doi.org/10.1090/S0025-5718-1977-0488722-3**[17]**R. A. Nicolaides,*On multigrid convergence in the indefinite case*, Math. Comp.**32**(1978), no. 144, 1082–1086. MR**0520340**, https://doi.org/10.1090/S0025-5718-1978-0520340-1**[18]**T. CRAIG POLING, M.A. Thesis, College of William and Mary, Williamsburg, Virginia, 1977.**[19]**R. V. Southwell,*Relaxation Methods in Theoretical Physics*, Oxford, at the Clarendon Press, 1946. MR**0018983**

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