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Fast solvers for finite difference approximations for the stokes and navier-stokes equations

Published online by Cambridge University Press:  17 February 2009

Dongho Shin  and
John C. Strikwerda
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Abstract

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We consider several methods for solving the linear equations arising from finite difference discretizations of the Stokes equations. The two best methods, one presented here for the first time, apparently, and a second, presented by Bramble and Pasciak, are shown to have computational effort that grows slowly with the number of grid points. The methods work with second-order accurate discretizations. Computational results are shown for both the Stokes equations and incompressible Navier-Stokes equations at low Reynolds number.

Type
Research Article
Information
The ANZIAM Journal , Volume 38 , Issue 2 , October 1996 , pp. 274 - 290
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Arrow, K., Hurwitz, L. and Uwaza, H., Studies in Nonlinear Programming (Stanford University Press, Stanford, 1958).Google Scholar
[2]Aziz, A. K. and Babuška, I., “Survey lectures on the mathematical foundations of the finite element mediod”, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (ed. Aziz, A. K.), (Academic Press, New York, 1972) 1–362.Google Scholar
[3]Bramble, J. H. and Pasciak, J. E., “A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems”, Math. Comp. 50 (1988) 1–18.Google Scholar
[4]Brandt, A. and Dinar, N., “Multigrid solutions to elliptic flow problems”, in Numerical Methods for Partial Differential Equations (ed. Parter, S. V.), (Academic Press, New York, 1979) 53–148.CrossRefGoogle Scholar
[5]Brandt, A., “Guide to multigrid development”, in Multigrid methods (eds. Hackbusch, W. and Trottenberg, U.), (Springer, New York, 1981) 220–312.Google Scholar
[6]Briggs, W. L., Multigrid Tutorial (Lancaster Press, Lancaster, Pennsylvania, 1987).Google Scholar
[7]Crozier, M., Approximation et méthodes iteratives de resolution d'inequations variationnelles et de problèmes non linéares, IRIA cahier no. 12 (1974).Google Scholar
[8]Fortin, M. and Glowinski, R., Resolution Numérique de Problèmes aux Limites par des Méthodes de Langrangien Augment (1981).Google Scholar
[9]Girault, V. and Raviart, P. A., “Finite Element Approximation of the Navier-Stokes Equations”, in: Lecture Notes in Mathematics 749 (Springer, New York, 1986).Google Scholar
[10]Hackbusch, W., Multi-Grid Methods and Applications (Springer, New York, 1985).Google Scholar
[11]Strikwerda, J.C., “Finite difference methods for the Stokes and Navier-Stokes equations”, SIAM J. Sci. Stat. Comput. 5 (1984) 56–68.CrossRefGoogle Scholar
[12]Strikwerda, J. C., “An iterative method for solving finite difference approximations to the Stokes equations”, SIAM J. Numer. Anal. 21 (1984) 447–58.Google Scholar
[13]Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations (Wadsworth and Books/Cole, Pacific Grove, CA, 1989).Google Scholar
[14]Strikwerda, J. C. and Scarbnick, C. D., “A domain decomposition method for incompressible viscous flow”, SIAM J. Sci. Stat. Comput. 14 (1993) 49–67.Google Scholar
[15]Wesseling, P., An Introduction To Multigrid Methods (John Wiley and Sons, New York, 1992).Google Scholar