Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A multigrid version of a simple finite element method for the Stokes problem


Authors: Juhani Pitkäranta and Tuomo Saarinen
Journal: Math. Comp. 45 (1985), 1-14
MSC: Primary 65N20; Secondary 65N30, 65N50, 76-08
DOI: https://doi.org/10.1090/S0025-5718-1985-0790640-2
MathSciNet review: 790640
Full-text PDF

Abstract | References | Similar Articles | 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.


References [Enhancements On Off] (What's this?)

  • [1] O. Axelsson, "Solution of linear systems of equations: iterative methods," in Sparse Matrix Techniques, Lecture Notes in Math., Vol. 572 (V. A. Barker, ed.), Springer-Verlag, Berlin and New York, 1977. MR 0448834 (56:7139)
  • [2] I. Babuška, "Error bounds for finite element methods," Numer. Math., v. 16, 1971, pp. 322-333. MR 0288971 (44:6166)
  • [3] I. Babuška, J. Osborn & J. Pitkäranta, "Analysis of mixed methods using mesh dependent norms," Math. Comp., v. 35, 1980, pp. 1039-1062. 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. 35-51. MR 595040 (82b:65113)
  • [5] F. Brezzi, "On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers," RAIRO Ser. Rouge, v. 8, no. R2, 1974, pp. 129-151. 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. 33-76. MR 0343661 (49:8401)
  • [7] V. Girault & P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Math., Vol. 749, Springer-Verlag, 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. 1-60. 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. 375-400. MR 645657 (83d:65287)
  • [11] J. Pitkäranta, "On a mixed finite element method for the Stokes problem in $ {{\mathbf{R}}^3}$," RAIRO Anal. Numér., v. 16, 1982, pp. 275-291.
  • [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 MAFELAP-1984 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), Springer-Verlag, 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)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N20, 65N30, 65N50, 76-08

Retrieve articles in all journals with MSC: 65N20, 65N30, 65N50, 76-08


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1985-0790640-2
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society