A multigrid version of a simple finite element method for the Stokes problem
Authors:
Juhani Pitkäranta and Tuomo Saarinen
Journal:
Math. Comp. 45 (1985), 114
MSC:
Primary 65N20; Secondary 65N30, 65N50, 7608
MathSciNet review:
790640
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Additional Information
Abstract: We consider a finite element method for the Stokes problem on a rectangular domain based on piecewise bilinear velocities and piecewise constant pressures on a uniform rectangular grid. It is shown that by a simple stabilization strategy the method can be implemented in a convergent multigrid procedure.
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 [3]
 I. Babuška, J. Osborn & J. Pitkäranta, "Analysis of mixed methods using mesh dependent norms," Math. Comp., v. 35, 1980, pp. 10391062. MR 583486 (81m:65166)
 [4]
 R. E. Bank & T. Dupont, "An optimal order process for solving elliptic finite element equations," Math. Comp., v. 36, 1981, pp. 3551. MR 595040 (82b:65113)
 [5]
 F. Brezzi, "On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrange multipliers," RAIRO Ser. Rouge, v. 8, no. R2, 1974, pp. 129151. MR 0365287 (51:1540)
 [6]
 M. Crouzeix & P.A. Raviart, "Conforming and nonconforming finite element methods for solving the stationary Stokes equations," RAIRO Ser. Rouge, v. 7, no. R3, 1973, pp. 3376. MR 0343661 (49:8401)
 [7]
 V. Girault & P.A. Raviart, Finite Element Approximation of the NavierStokes Equations, Lecture Notes in Math., Vol. 749, SpringerVerlag, Berlin and New York, 1979. MR 548867 (83b:65122)
 [8]
 W. Hackbusch, Analysis and Multigrid Solutions of Mixed Finite Element and Mixed Difference Equations, Preprint, Universität Bochum, 1980.
 [9]
 T. J. Hughes, W. K. Liu & A. Brooks, "Finite element analysis of incompressible viscous flows by the penalty function formulation," J. Comput. Phys., v. 30, 1979, pp. 160. MR 524162 (80b:76008)
 [10]
 C. Johnson & J. Pitkäranta, "Analysis of some mixed finite element methods related to reduced integration," Math. Comp., v. 38, 1982, pp. 375400. MR 645657 (83d:65287)
 [11]
 J. Pitkäranta, "On a mixed finite element method for the Stokes problem in ," RAIRO Anal. Numér., v. 16, 1982, pp. 275291.
 [12]
 J. Pitkäranta & R. Stenberg, Error Bounds for the Approximation of the Stokes Problem Using Bilinear/Constant Elements on Irregular Quadrilateral Meshes, Proceedings of the MAFELAP1984 conference. (To appear.)
 [13]
 K. Stüben & U. Trottenberg, "Multigrid methods: fundamental algorithms, model problem analysis and applications," in Multigrid Methods, Lecture Notes in Math., Vol. 960 (W. Hackbusch & U. Trottenberg, eds), SpringerVerlag, Berlin and New York, 1982. MR 685773 (84m:65129)
 [14]
 R. Verfürth, A Multilevel Algorithm for Mixed Problems, Preprint, Universität Bochum, 1982.
 [15]
 J. M. Boland & R. A. Nicolaides, Stable and Semistable Low Order Finite Elements for Viscous Flows, Preprint, University of Connecticut, 1983. MR 787571 (86m:65139)
 [16]
 A. Brandt & N. Dinar, "Multigrid solutions to elliptic flow problems," in Numerical Methods for Partial Differential Equations (S. V. Parter, ed.), Academic Press, New York, 1979. MR 558216 (81a:65094)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507906402
PII:
S 00255718(1985)07906402
Article copyright:
© Copyright 1985
American Mathematical Society
