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

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- Math. Comp.
**42**(1984), 505-519 Request permission

## 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.## References

- Randolph E. Bank and Todd Dupont,
*An optimal order process for solving finite element equations*, Math. Comp.**36**(1981), no. 153, 35–51. MR**595040**, DOI 10.1090/S0025-5718-1981-0595040-2
R. E. Bank & T. Dupont, - Dietrich Braess,
*The contraction number of a multigrid method for solving the Poisson equation*, Numer. Math.**37**(1981), no. 3, 387–404. MR**627112**, DOI 10.1007/BF01400317 - Achi Brandt,
*Multi-level adaptive solutions to boundary-value problems*, Math. Comp.**31**(1977), no. 138, 333–390. MR**431719**, DOI 10.1090/S0025-5718-1977-0431719-X
A. Brandt, private communication.
- Achi Brandt and Nathan Dinar,
*Multigrid solutions to elliptic flow problems*, Numerical methods for partial differential equations (Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 42, Academic Press, New York-London, 1979, pp. 53–147. MR**558216**
H. Foerster, K. Stüben & U. Trottenberg, "Non-standard multigrid techniques using checkered relaxation and intermediate grids," in - Wolfgang Hackbusch,
*Survey of convergence proofs for multigrid iterations*, Special topics of applied mathematics (Proc. Sem., Ges. Math. Datenverarb., Bonn, 1979) North-Holland, Amsterdam-New York, 1980, pp. 151–164. MR**585154** - S. F. McCormick and J. W. Ruge,
*Multigrid methods for variational problems*, SIAM J. Numer. Anal.**19**(1982), no. 5, 924–929. MR**672568**, DOI 10.1137/0719067 - T. Meis and H.-W. Branca,
*Schnelle Lösung von Randwertaufgaben*, Z. Angew. Math. Mech.**62**(1982), no. 5, T263–T270. MR**677307**, DOI 10.1002/zamm.19820620504 - W. L. Miranker and V. Ya. Pan,
*Methods of aggregation*, Linear Algebra Appl.**29**(1980), 231–257. MR**562764**, DOI 10.1016/0024-3795(80)90245-1 - R. A. Nicolaides,
*On the $l^{2}$ convergence of an algorithm for solving finite element equations*, Math. Comp.**31**(1977), no. 140, 892–906. MR**488722**, DOI 10.1090/S0025-5718-1977-0488722-3 - R. A. Nicolaides,
*On some theoretical and practical aspects of multigrid methods*, Math. Comp.**33**(1979), no. 147, 933–952. MR**528048**, DOI 10.1090/S0025-5718-1979-0528048-4 - Joachim A. Nitsche and Alfred H. Schatz,
*Interior estimates for Ritz-Galerkin methods*, Math. Comp.**28**(1974), 937–958. MR**373325**, DOI 10.1090/S0025-5718-1974-0373325-9 - Manfred Ries, Ulrich Trottenberg, and Gerd Winter,
*A note on MGR methods*, Linear Algebra Appl.**49**(1983), 1–26. MR**688373**, DOI 10.1016/0024-3795(83)90091-5 - Richard S. Varga,
*Matrix iterative analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR**0158502**
R. Verfürth, "The contraction number of a multigrid method with mesh ratio 2 for solving Poisson’s equation," - D. Braess and W. Hackbusch,
*A new convergence proof for the multigrid method including the $V$-cycle*, SIAM J. Numer. Anal.**20**(1983), no. 5, 967–975. MR**714691**, DOI 10.1137/0720066

*Analysis of a Two-Level Scheme for Solving Finite Element Equations*, Report CNA-159, Center for Numerical Analysis, Austin, 1980.

*Elliptic Problem Solvers*(M. Schultz, ed.), Academic Press, New York, 1981. W. Hackbusch, "On the convergence of multi-grid iterations,"

*Beiträge Numer. Math.*, v. 9, 1981, pp. 213-239.

*Linear Algebra Appl.*(To appear.)

## Additional Information

- © Copyright 1984 American Mathematical Society
- Journal: Math. Comp.
**42**(1984), 505-519 - MSC: Primary 65N20
- DOI: https://doi.org/10.1090/S0025-5718-1984-0736449-6
- MathSciNet review: 736449