Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

Grad-div stablilization for Stokes equations


Authors: Maxim A. Olshanskii and Arnold Reusken
Translated by:
Journal: Math. Comp. 73 (2004), 1699-1718
MSC (2000): Primary 65N30, 65N22, 76D07
DOI: https://doi.org/10.1090/S0025-5718-03-01629-6
Published electronically: December 19, 2003
MathSciNet review: 2059732
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper a stabilizing augmented Lagrangian technique for the Stokes equations is studied. The method is consistent and hence does not change the continuous solution. We show that this stabilization improves the well-posedness of the continuous problem for small values of the viscosity coefficient. We analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact Uzawa method.


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

  • 1. BANK, R.E., WELFERT, B.D., AND YSERENTANT, H. A class of iterative methods for solving saddle point problems, Numer. Math., 56 (1990), pp. 645-666.MR 91b:65035
  • 2. BRAMBLE, J.H., AND PASCIAK, J.E. Iterative techniques for time dependent Stokes problems, Comput. Math. Appl. 33 (1997), pp. 13-30.MR 98e:65091
  • 3. BRAMBLE, J.H., PASCIAK, J.E., AND VASSILEV, A.T. Analysis of the inexact Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 33 (1997) , pp. 1072-1092.MR 98c:65182
  • 4. BYCHENKOV, YU.V. AND CHIZONKOV, E.V. Optimization of one three-parameter method of solving an algebraic system of Stokes type, Russ. J. Numer. Anal. Math. Model, 14 (1999), pp. 429-440.MR 2000h:65062
  • 5. CAHOUET, J., AND CHABARD, J.P. Some fast 3D finite element solvers for the generalized Stokes problem, Int. J. Numer. Methods Fluids., 8 (1988), pp. 869-895.MR 89i:76005
  • 6. CHIZHONKOV, E.V., AND OLSHANSKII, M.A., On the domain geometry dependence of the LBB condition, Math. Model. Anal. Numer. (M2AN), 34 (2000) pp. 935-951. MR 2002c:65203
  • 7. FUJITA, H, AND SUZUKI, T. Evolution problems, in: Handbook of numerical analysis (eds. P.G. Ciarlet and J.L. Lions), Vol. 2, pp. 789-928, North-Holland, Amsterdam, 1991.
  • 8. GIRAULT, V., AND RAVIART, P.-A. Finite element methods for Navier-Stokes equations, Springer, Berlin, 1986.MR 88b:65129
  • 9. GLOWINSKI, R., AND LE TALLEC, P. Augmented lagrangian and operator splitting methods in nonlinear mechanics, SIAM, Studies in Applied Mathematics, Philadelphia, 1989.MR 91f:73038
  • 10. KOBELKOV G.M. On solving the Navier-Stokes equations at large Reynolds numbers, Russ. J. Numer. Anal. Math. Model, 10 (1995), pp. 33-40.MR 95m:65167
  • 11. KOBELKOV, G.M. AND OLSHANSKII, M.A. Effective preconditioning of Uzawa type schemes for a generalized Stokes problem, Numer. Math., 86 (2000), pp. 443-470.MR 2001j:65168
  • 12. OLSHANSKII, M.A. A low order Galerkin finite element method for the Navier-Stokes equations of steady incompressible flow: a stabilization issue and iterative methods, Comput. Meth. Appl. Mech. Eng., 191 (2002), pp. 5515-5536.
  • 13. OLSHANSKII, M.A. AND REUSKEN, A. On the convergence of a multigrid algorithm for linear reaction-diffusion problems, Computing, 65 (2000), pp. 193-202. MR 2001m:65186
  • 14. QUARTERONI, A. AND VALLI, A. Numerical approximation of partial differential equations Springer, Berlin 1997.MR 95i:65005
  • 15. ROOS, H.-G., STYNES, M., AND TOBISKA, L. Numerical methods for singularly perturbed differential equations: convection diffusion and flow problems Springer Series in Computational Mathematics, 24. Springer-Verlag, Berlin, 1996.MR 99a:65134
  • 16. RUSTEN, T., AND WINTHER, R. A preconditioned iterative method for saddlepoint problems SIAM J. Matrix Anal. Appl. 13 (1992), pp. 887-904.MR 93a:65043
  • 17. SCHÖBERL, J. Multigrid methods for a parameter dependent problem in primal variables, Numer. Math., 84 (1999), pp. 97-119.MR 2001a:65147
  • 18. SILVESTER, D., AND WATHEN, A. Fast iterative solution of stabilised Stokes systems. Part II: using general block preconditioners, SIAM J. Numer. Anal., 31 (1994), pp. 1352-1367. MR 95g:65132
  • 19. THOMEE, V. Galerkin finite element methods for parabolic problems, Springer, Berlin, 1984.MR 86k:65006
  • 20. TOBISKA, L., AND VERF¨URTH, R. Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations, SIAM J. Numer. Anal., 33 (1996), pp.107-127.MR 97e:65133
  • 21. VASSILEVSKI, P.S., AND LAZAROV, R.D. Preconditioning mixed finite element saddle-point elliptic problems, Numer. Linear Alg. Appl., 3 (1996) , pp. 1-20.MR 96m:65111
  • 22. VERFÜRTH, R. A combined conjugate gradient-multigrid algorithm for the numerical solution of the Stokes problem IMA J. Numer. Anal. 4 (1984), pp. 441-455.MR 86f:65200
  • 23. ZULEHNER, W. Analysis of iterative methods for saddle point problems: A unified approach, Math. Comp. 71 (2001), pp. 479-505. MR 2003f:65183

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 65N22, 76D07

Retrieve articles in all journals with MSC (2000): 65N30, 65N22, 76D07


Additional Information

Maxim A. Olshanskii
Affiliation: Dept. Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia
Email: ay@olshan.msk.ru

Arnold Reusken
Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH-Aachen, D-52056 Aachen, Germany
Email: reusken@igpm.rwth-aachen.de

DOI: https://doi.org/10.1090/S0025-5718-03-01629-6
Keywords: Stokes equations, finite elements, augmented Lagrangian, inexact Uzawa
Received by editor(s): November 7, 2001
Received by editor(s) in revised form: March 5, 2003
Published electronically: December 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society