# Completely splitting method for the Navier-Stokes problem

• I. V. Kireev
• U. Rüde
• V. V. Shaidurov
Conference paper
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM) book series (NNFM, volume 88)

## Summary

We consider two-dimensional time-dependent Navier-Stokes equations in a rectangular domain and study the method of full splitting [3]–[4]. On the physical level, this problem is splitted into two processes: convection-diffusion and action of pressure. The convection-diffusion step is further splitted in two geometric directions. To implement the finite element method, we use the approach with uniform square grids which are staggered relative to one another. This allows the Ladyzhenskaya-Babuška-Brezzi condition for stability of pressure to be fulfilled without usual diminishing the number of degrees of freedom for pressure relative to that for velocities. For pressure we take piecewise constant finite elements. As for velocities, we use piecewise bilinear elements.

## Keywords

Linear Algebraic Equation Multigrid Method Split Method Fractional Step Extended Domain
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## References

1. 1.
Boland J, Nicolaides R (1983) SIAM J Numer Math 20:722–731
2. 2.
Brezzi F, Fortin M (1991) Mixed and Hybrid Finite Element Metods. Springer, New YorkGoogle Scholar
3. 3.
Chorin A (1967) J Comp Phys 2:12–26
4. 4.
Chorin A (1969) Math Comp 23:341–353
5. 5.
Glowinski R (1987) Le Θ-scheme. In: Bristian M, Glowinski R, Perinx J (eds) Numerical methods for Navier-Stokes equationsGoogle Scholar
6. 6.
Hackbusch W (1985) Multigrid Methods and Applications. Springer, BerlinGoogle Scholar
7. 7.
Heywood J, Rannacher R (1982) SIAM J Numer Anal 19:275–311
8. 8.
Heywood J, Rannacher R (1990) SIAM J Numer Anal 17:353–384
9. 9.
Kloucek P, Rys F (1994) SIAM J Numer Anal 31:1312–1336
10. 10.
Mansfield L (1984) Mumer Math 45:165–172
11. 11.
Marchuk G, Shaidurov V (1983) Difference Methods and Their Extrapolations. Springer, New York
12. 12.
Prohl A (1997) Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations. Teubrer, Studtgart
13. 13.
Rannacher R (2000) Finite Element Methods for the Incompressible Navier-Stokes Equations. In: Galdi G, Heywood J, Rannacher R (eds) Fundamental Directions in Mathematical Fluid Mechanics. Birkhäuser, BerlinGoogle Scholar
14. 14.
Rüde U (1994) Multilevel, extrapolation, and sparse grid methods. In: Proceedings of the Fourth European Conference on Multigrid Methods. BostonGoogle Scholar
15. 15.
Tirek S (1999) Efficient Solvers for Incompressible Flow Problems. Springer, Berlin HeidelbergGoogle Scholar
16. 16.
Shaidurov V (1995) Multigrid Methods for Finite Elements. Kluwer Academic Publishers, Dordrecht
17. 17.
Shen J (1996) Math Comp 65:1039–1065
18. 18.
Temam R (1979) Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland Publishing Company, AmsterdamGoogle Scholar
19. 19.
Van Kan (1986) SIAM J Sci Staf Comp 7:870–891

## Authors and Affiliations

• I. V. Kireev
• 1
• U. Rüde
• 2
• V. V. Shaidurov
• 1
1. 1.Institute of Computational Modelling SB RASAcademgorodokKrasnoyarskRussia
2. 2.University of Erlangen-NurembergErlangenGermany