A modified finite element method for solving the time-dependent, incompressible Navier-Stokes equations. I: Theory. (English) Zbl 0559.76030

Beginning with the Galerkin finite element method and the simplest appropriate isoparametric element for modelling the Navier-Stokes equations, the spatial approximation is modified in two ways in the interest of cost-effectiveness: the mass matrix is ”lumped” and all coefficient matrices are generated via 1-point quadrature. After appending an hour-glass correction term to the diffusion matrices, the modified semi-discretized equations are integrated in time using the forward explicit Euler method in a special way to compensate for that portion of the time truncation error which is intolerable for advection- dominated flows. The scheme is completed by the introduction of a subcycling strategy that permits less frequent updates of the pressure field with little loss of accuracy.


76D05 Navier-Stokes equations for incompressible viscous fluids
76M99 Basic methods in fluid mechanics


Zbl 0559.76031
Full Text: DOI


[1] and , ’On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions’, in Recent Advances in Numerical Methods in Fluids, vol. 1, Pineridge Press, Swansea, U. K., 1980, p. 27.
[2] Engelman, Adv. Eng. Software 4 pp 163– (1982)
[3] Flanagan, Int. J. Num. Meth. Eng. 17 pp 679– (1981)
[4] Petzold, SIAM J. Sci. Stat. Comput. 3 pp 367– (1982)
[5] Gresho, Int. J. Num. Meth. Eng. 18 pp 1260– (1982)
[6] Sani, Int. J. Num. Meth. Fluids 1 pp 17– (1981)
[7] Gresho, Comp. Fluids 9 pp 223– (1981)
[8] ’The finite element method’, in Numerical Methods Used in Atmospheric Models, Vol II, GARP Publication Series No. 17, World Meteorological Organization, 1979.
[9] Leone, Int. J. Num. Meth. Eng. 14 pp 769– (1979)
[10] Brooks, Comp. Meth. Appl. Mech. Eng. 32 pp 199– (1982)
[11] The Finite Element Method, McGraw-Hill, London, 1977.
[12] Lee, Int. J. Num. Meth. Eng. 14 pp 1785– (1979)
[13] Cullen, J. Comp. Phys. 51 pp 273– (1983)
[14] and , ’Further studies on equal-order interpolation for Navier-Stokes’, Fifth Int. Sym. on Finite Elements in Flow Problems, Proceedings, Austin, Texas, 23-23 January 1984; also UCRL-89094.
[15] personal communication, 1981.
[16] Trefethen, SIAM Review 24 pp 113– (1982)
[17] and , Numerical Weather Prediction and Dynamic Meteorology, Wiley, New York, 1980, p. 477.
[18] Smolarkiewicz, Monthly Weather Review 110 pp 1968– (1982)
[19] Kosloff, Int. J. Num. Anal. Meth. Geomech. 2 pp 57– (1978)
[20] ’Users manual for DYNA2D–an explicit two-dimensional hydrodynamic finite element code with interactive rezoning’, Lawrence Livermore National Laboratory Report UCID-18756, 1980.
[21] Goudreau, Comp. Meth. Appl. Mech. Eng. 33 pp 1– (1982)
[22] and , ’Direct solution of equations by forntal and variable band, active column methods’, in Nonlinear Finite Element Analysis in Structural Mechanics, Springer-Verlag, 1981, 521.
[23] and , ’The stability of explicit Euler time integration for certain finite difference approximations of the multi-dimensional advection-diffusion equation’, Int. J. Num. Meth. Fluids, to be published. Also available as Lawrence Livermore National Laboratory Report UCRL-88519.
[24] Dukowicz, J. Comp. Phys. 32 pp 71– (1979)
[25] and , Numerical Solution of Partial Differential Equations, Wiley, New York, 1982.
[26] Computational Fluid Dynamics, Hermosa Publishers, P. O. Box 8172, Albuquerque, N. M., 1976.
[27] Lax, Comm. Pure & Appl. Math. 17 pp 381– (1964)
[28] Orzag, J. Fl. Mech. 49 pp 75– (1971)
[29] and , ’Advection-dominated flows, with emphasis on the consequences of mass lumping’, in Finite Elements in Fluids– Vol. 3, Wiley, Chichester, 1978, Chap. 19, p. 335.
[30] Roache, J. Comp. Phys. 10 pp 169– (1972)
[31] de Vahl Davis, Comp. fluids 4 pp 29– (1976)
[32] Raithby, Comp. Meth. Appl. Mech. Eng. 9 pp 153– (1976)
[33] Leone, J. Comp. Phys. 41 pp 167– (1981)
[34] Buoyancy Effects in Fluids, Cambridge University Press, Cambridge, 1973. · Zbl 0262.76067
[35] and , ’Current progress in solving the time-dependent, incompressible Navier-Stokes equations in three dimensions by (almost) the FEM’, Proceedings, Fourth Int. Conf. on Finite Elements in Water Resources, Hanover, Germany, 21-21 June 1982. Also, UCRL-87445.
[36] Sun, Monthly Weather Review 108 pp 402– (1980)
[37] and , ’Pressure methods for the approximate solution of the Navier-Stokes equations’, Num. Meth. in Laminar & Turbulent Flow, Proc., 3rd Int. Conf., Seattle, 8-8 August 1983.
[38] , and , ’The imposition of incompressibility constraints via variational adjustment of velocity fields’, in Proceedings, Num. Methods for Laminar and Turbulent Flow, Pentech Press, London, 1978, p. 983. Also Lawrence Livermore National Laboratory Report UCRL-80553.
[39] Sherman, J. Appl. Met. 17 pp 312– (1976)
[40] and , ’Application of a modified finite element method to the time-dependent thermal convection of liquid metal’, Proceedings, Int. Conf. Num. Meth. Laminar and Turbulent Flow, University of Washington, 8-8 August 1983. Also Lawrence Livermore Laboratory Report UCRL-88990.
[41] Applied Linear Algebra, Prentice-Hall, N. J., 1969.
[42] and , A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, N. Y., 1979. · Zbl 0417.76002
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