An Iterative Solver for the Oseen Problem and Numerical Solution of Incompressible Navier-Stokes Equations

Numerical Linear Algebra with Applications

Volumn 6, Issue 5, 1999, Pages 353-378

  Olshanskii, Maxim A ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

Convection diffusion; LATEX; Multigrid; Navier Stokes; Oseen problem; Schur complement; Singular perturbed problem

Indexed keywords

EID: 0033436754     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1099-1506(199907/08)6:5<353::AID-NLA169>3.0.CO;2-J     Document Type: Article

References (37)

1. K. Arrow, L. Hurwicz and H. Uzawa. Studies in Nonlinear Programming. Stanford University Press, Stanford, 1958.
2. O. Axelsson. Iterative Solution methods. Cambridge University Press, Cambridge, 1994.
3. R. Bank, B. Welfert and H. Yserentant. A class of iterative methods for solving saddle point problems. Numer. Math., 56, 645-666, 1990.
4. J. Bramble and J. Pasciak. A preconditioning technique for indefinite systems resulting from mixed approximation of elliptic problems. Math. Comp., 50, 1-17, 1988.
5. J. Bramble and J. Pasciak. Iterative techniques for time-dependent Stokes problem. Comput. Math. Appl., 33, 13-30, 1997.
6. J. Bramble, J. Pasciak and A. Vassilev. Inexact Uzawa algorithms for nonsymmetric saddle point problems. Math. Comp., (to appear).
7. J. Bramble, J. Pasciak and A. Vassilev. Analysis of the inexact Uzawa algorithm for saddle point problems. SIAM J. Numer. Anal., 33, 1072-1092, 1997.
8. F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, 1991.
9. J. Cahouet and J. Chabart. Some fast 3D finite element solvers for the generalized Stokes problem. Int. J. Numer. Methods Fluids, 8, 869-895, 1988.
10. E. Chizhonkov. On convergence of the artificial compressibility method. Vestnik Mosk. Univ.,Ser.1., 2, 13-20, 1996.
11. E. D'yakonov. Minimization of Computational Work. Asymptotically Optimal Algorithms for Elliptic Problems. Nauka, Moscow, 1989.
12. H. Elman. Preconditioning for the steady-state Navier-Stokes equations with low viscosity. SIAM J. Sci. Comput., 20, 1299-1316, 1999.
13. H. Elman and G. Golub. Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal., 31, 1645-1661, 1994.
14. H. Elman and D. Silvester. Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations. SIAM J. Sci. Comput., 17, 33-46, 1996.
15. V. Girault and P. Raviart. Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, Berlin, 1986.
16. R. Glowinski. Finite element methods for the numerical simulations of incompressible viscous flow. Introduction to the control of the Navier-Stokes equations. In Lectures in Applied Mathematics, volume 28, pages 219-301. AMS, Providence, RI, 1991.
17. G. Golub and A. Wathen. An iteration for indefinite systems and its application for the Navier-Stokes equations. SIAM J. Sci. Comput., 19, 530-539, 1998.
18. W. Hackbush. Multi-grid Methods and Applications. Springer-Verlag, Berlin, 1985.
19. J. Heywood and R. Rannacher. Finite element approximation ot the nonstationary Navier-Stokes problem, part iv: error analysis for second-order time discretization. SIAM J. Numer. Anal., 27, 353-358, 1990.
20. C. Johnson, R. Rannacher and M. Boman. Numerics and hydrodynamic stability: towards error control in computation fluid dynamics. SIAM J. Numer. Anal., 32, 1058-1079, 1995.
21. A. Klawonn and G. Starke. Block triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis. Numer. Math., 81, 577-594, 1999.
22. G. Kobelkov. On numerical methods of solving the Navier-Stokes equations in velocity-pressure variables. In C.Marchuk, editor, Numerical Methods and Applications. CRC Press, New York, 1994.
23. G. Kobelkov and M. Olshanskii. Effective preconditioning of Uzawa type schemes for generalized Stokes problem. Technical Report 9718, Department of Mathematics University of Nijmegen, June 1997. (accepted to Numer. Math.).
24. L. A. Krukier. Special preconditioners for the iterative solution of strongly nonsymmetric linear systems. In B. O.Axelsson, M.Neytcheva, editors, Proceedings of the Conference on Preconditioning Itererative Solution Methods in Large Scale Problems in Scientific Computing, Nijmegen, The Netherlands, 1997.
25. O. Ladyzhenskaya. Boundary Problems of Mathematical Physics. Nauka, Moscow, 1973.
26. L. Landau and E. Lifshits. Fluid Mechanics Pergamon. New York, 1975.
27. M. Olshanskii. On numerical solution of nonstationary Stokes equations. Russ. J. Numer. Anal. Math. Modelling, 10(10), 81-92, 1995.
28. M. Olshanskii. Stokes problem with model boundary conditions. Sbornik: Mathematics, 188, 603-620, 1997.
29. O. Pironneau. Finite Element Methods for Fluids. Wiley, Masson, Paris, 1989.
30. W. Queck. The convergence factor of preconditioned algorithm of the Arrow-Hurwicz type. SIAM J. Numer. Anal., 26, 1016-1030, 1989.
31. T. Rusten and R. Winther. A preconditioned iterative method for saddle point problems. SIAM J. Matrix Anal. Appl., 13, 887-904, 1992.
32. D. Silvester and A. Wathen. Fast iterative solution of stabilized Stokes systems part ii: using block preconditioners. SIAM J. Numer. Anal., 31, 1352-1367, 1994.
33. R. Temam. Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam, 1977.
34. S. Turek. A comparative study of time stepping techniques for the incompressible Navier-Stokes equations: from fully implicit non-linear schemes to semi-implicit projection methods. Int. J. Numer. Methods Fluids, 22, 987-1011, 1996.
35. S. Turek. Efficient Solvers for Incompressible Flow Problems: An Algorithmic Approach in View of Computational Aspects, volume 6 of Lecture Notes on Computer Science in Engineering. Springer-Verlag, Heidelberg, 1999.
36. A. Wathen. Realistic eigenvalues bounds for the Galerkin mass matrix. IMA J. Numer. Anal., 7, 449-457, 1987.
37. A. Wathen and D. Silvester. Fast iterative solution of stabilized Stokes systems part i: using simple diagonal preconditioners. SIAM J. Numer. Anal., 30,630-649, 1993.