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The dual variable method for finite element discretizations of Navier/Stokes equations.(English)Zbl 0587.76045

The numerical solution of two-dimensional, transient, incompressible Navier-Stokes problems is considered. The dual variable method, originally developed in the context of a finite difference discretization, is a technique to considerably reduce the size of the linear system to be solved at each time step. The steps involved in the method are (1) the determination of the rank of the discrete divergence operator, A, (2) the determination of a basis for the null space of A, N(A), and (3) the calculation of a particular solution of the discrete continuity equation. A finite element implementation of the method is presented using quadrilateral piecewise bilinear velocity/constant pressure elements. Algorithms for the determination of a basis for N(A) and a particular solution are presented. Numerical comparisons of primitive versus dual variable formulations on several problems demonstrate the advantage of the dual variable method, in terms of both execution speed and memory requirements.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76M99 Basic methods in fluid mechanics
65N06 Finite difference methods for boundary value problems involving PDEs
35Q30 Navier-Stokes equations

Software:

LINPACK

Cite

References:

[1]Amit, J. Comp. Phys. 40 pp 183– (1981)
[2]and , Programming. Games and Transporation, Methuen, London, 1965.
[3]and , ’On the Kizhivitski-Ladyzhenskaya difference method for the stationary Navier-Stokes equations’, Proc. 10th IMACS World Congress, Montreal, Canada (Ed.), 1982.
[4], and , The LINPACK User’s Guide, SIAM, Philadelphia, PA, 1979. ·doi:10.1137/1.9781611971811
[5]Griffiths, Int. j. numer. methods fluids 1 pp 323– (1981)
[6]and , ’Numerical methods for thermally expandable two-phase flow–computational techniques for steam generator modelling’, Electric Power Research Institute, Report No. EPRI NP-1416, (May 1980).
[7]Hecht, Serie Analyse Numerique 15 pp 119– (1981)
[8]Hood, Comp. Fluids 1 pp 73– (1973)
[9]’On eigenvalue approximations by mixed finite element methods’, Ph.D. dissert., Univ. of Tennessee, Knoxville, Tennessee (1980).
[10]Porsching, Nucl. Sci. Eng. 64 pp 177– (1977)
[11]and , ’On the spurious pressures generated by certain GFEM solutions of the incompressible Navier-Stokes equations’, Proc. Third Int. Conf. on Finite Elements in Flow Problems, Baniff, Alberta, Canada (1980). ·Zbl 0446.76034
[12]’A finite element implementation of the dual variable method for the Navier Stokes’. Ph.D. Dissertation, Univ. of Pittsburgh, Pittsburgh PA (1983).
[13]Linear Algebra and Its Applications, Academic Press, New York, 1976.
[14]The Finite Element Method in Engineering Science, McGraw-Hill, London, 1971.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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