The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation

Author:
Dietrich Braess

Journal:
Math. Comp. **42** (1984), 505-519

MSC:
Primary 65N20

DOI:
https://doi.org/10.1090/S0025-5718-1984-0736449-6

MathSciNet review:
736449

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Abstract: The convergence rate of a multigrid method for the numerical solution of the Poisson equation on a uniform grid is estimated. The results are independent of the shape of the domain as long as it is convex and polygonal. On the other hand, pollution effects become apparent when the domain contains reentrant corners. To estimate the smoothing of the Gauss-Seidel relaxation, the smoothness is measured by comparing the energy norm with a (weaker) discrete seminorm.

**[1]**1. R. E. Bank & T. Dupont, "An optimal order process for solving finite element equations,"*Math. Comp.*, v. 36, 1981, pp. 35-51. MR**595040 (82b:65113)****[2]**R. E. Bank & T. Dupont,*Analysis of a Two-Level Scheme for Solving Finite Element Equations*, Report CNA-159, Center for Numerical Analysis, Austin, 1980.**[3]**D. Braess, "The contraction number of a multigrid method for solving the Poisson equation,"*Numer. Math.*, v. 37, 1981, pp. 387-404. MR**627112 (82h:65073)****[4]**A. Brandt, "Multi-level adaptive solutions to boundary-value problems,"*Math. Comp.*, v. 31, 1977, pp. 333-390. MR**0431719 (55:4714)****[5]**A. Brandt, private communication.**[6]**A. Brandt & N. Dinar, "Multigrid solutions to elliptic flow problems," in*Numerical Methods for PDEs*(S. Parter, ed.), Academic Press, New York, 1979, pp. 53-107. MR**558216 (81a:65094)****[7]**H. Foerster, K. Stüben & U. Trottenberg, "Non-standard multigrid techniques using checkered relaxation and intermediate grids," in*Elliptic Problem Solvers*(M. Schultz, ed.), Academic Press, New York, 1981.**[8]**W. Hackbusch, "On the convergence of multi-grid iterations,"*Beiträge Numer. Math.*, v. 9, 1981, pp. 213-239.**[9]**W. Hackbusch, "Survey of convergence proofs for multi-grid iterations," in*Special Topics of Applied Mathematics*(J. Frehse, D. Pallaschke, U. Trottenberg, eds.), North-Holland, Amsterdam, 1980, pp. 152-164. MR**585154 (82j:65072)****[10]**S. F. McCormick & J. W. Ruge, "Multigrid methods for variational problems,"*SIAM J. Numer. Anal.*, v. 19, 1982, pp. 924-929. MR**672568 (84g:49049)****[11]**Th. Meis & H.-W. Branca, "Schnelle Lösung von Randwertaufgaben,"*ZAMM*, v. 62, 1982, pp. T263-T270. MR**677307 (84e:65108)****[12]**W. L. Miranker & V. Ya. Pan, "Methods of aggregation,"*Linear Algebra Appl.*, v. 29, 1980, pp. 231-257. MR**562764 (82d:65036)****[13]**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)****[14]**R. A. Nicolaides, "On some theoretical and practical aspects of multigrid methods,"*Math. Comp.*, v. 33, 1979, pp. 933-952. MR**528048 (80c:65209)****[15]**J. A. Nitsche & A. H. Schatz, "Interior estimates for Ritz-Galerkin methods,"*Math. Comp.*, v. 28, 1974, pp. 937-958. MR**0373325 (51:9525)****[16]**M. Ries, U. Trottenberg & G. Winter, "A note on MGR methods,"*Linear Algebra Appl.*, v. 49, 1983, pp. 1-26, supplemented by private communication of the second author. MR**688373 (84f:65030)****[17]**R. S. Varga,*Matrix Iterative Analysis*, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR**0158502 (28:1725)****[18]**R. Verfürth, "The contraction number of a multigrid method with mesh ratio 2 for solving Poisson's equation,"*Linear Algebra Appl.*(To appear.)**[19]**D. Braess & W. Hackbusch, "A new convergence proof for the multigrid method including the*V*-cycle,"*SIAM J. Numer. Anal.*, v. 20, 1983, pp. 967-975. MR**714691 (85h:65233)**

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DOI:
https://doi.org/10.1090/S0025-5718-1984-0736449-6

Article copyright:
© Copyright 1984
American Mathematical Society