On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. II: Implementation. (English) Zbl 0712.76036

Summary: [For part I, see the foregoing entry (Zbl 0712.76035).]
Ever since the expansion of the finite element method (FEM) into unsteady fluid mechanics, the ‘consistent mass matrix’ has been a relevant issue. Applied to the time-dependent incompressible Navier-Stokes equations, it virtually demands the use of implicit time integration methods in which full ‘velocity-pressure coupling’ is also inherent. The high cost of such (high-quality) FEM calculations led to the development of simpler but ad hoc methods in which the ‘lumped’ mass matrix is employed and the velocity and pressure are uncoupled to the maximum extent possible. Resulting computer codes were less expensive to use but suffered a significant loss of accuracy, caused by lumping the mass when the flow was advection-dominated and accurate transport of ‘information’ was important. In the second part of this paper we re-introduce the consistent mass matrix into some semi-implicit projection methods in such a way that the cost advantage of lumped mass and the accuracy advantage of consistent mass are simultaneously realized.


76D05 Navier-Stokes equations for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics


Zbl 0712.76035
Full Text: DOI


[1] Gresho, Int. j. numer. methods fluids 4 pp 557– (1984)
[2] and , ’A new semi-implicit method for solving the time-dependent conservation equations for incompressible flow’, Proc. Numerical Methods in Laminar and Turbulent Flow, Pineridge Press, Swansea, 1985, pp. 3-21.
[3] and , ’Advection-dominated flows, with emphasis on the consequences of mass lumping’, Proc. Second Int. Symp. on Finite Element Methods in Flow Problems, Int. Center for Computer-Aided Design, Santa Margherita, Italy, 1976, pp. 743-756.
[4] and , ’Advection-dominated flows, with emphasis on the consequences of mass lumping’, Finite Elements in Fluids, Vol. 3, Wiley, Chichester, 1978, pp. 335-350.
[5] Treffethen, SIAM Rev. 24 pp 113– (1982)
[6] and , ’A comparative study of certain finite-element and finite-difference methods in advection-diffusion simulations’, Proc. Summer Computer Simulation Conf., Washington, DC, 12-14 July 1976, pp. 37-42.
[7] and , ’Some new results using quadratic finite elements for pure advection’, Proc. IMACS Symp. on Numerical Solutions of PDE’s, Int. Center for Computer-Aided Design, Leheigh University, June 1987, pp. 202-209.
[8] Gresho, Adv. Water Resources 4 pp 175– (1981)
[9] Gresho, Comput. Fluids 9 pp 223– (1981)
[10] and , ’Solving the incompressible Navier-Stokes equations using consistent mass and a pressure Poisson equation’, Proc. ASME Symp. on Recent Advances in Computational Fluid Dynamics, Chicago, IL, 28 November-2 December 1988, pp. 51-75.
[11] and , ’Semi-consistent mass matrix techniques for solving the incompressible Navier-Stokes equations’, LLNL Report UCRL-99503, 1988.
[12] and , ’On the time-dependent solution of the incompressible Navier-Stokes equations in two- and three-dimensions’, Recent Advances in Numerical Methods in Fluids, Vol. 1, Pineridge Press, Swansea, 1980, pp. 27-81.
[13] Petzold, SIAM J. Sci. Stat. Comput. 7 pp 720– (1986)
[14] and , Incompressible Flows and the Finite Element Method, Wiley, Chichester, 1991 (in preparation).
[15] , , and , ’Conservation laws for primitive variable formulations of the incompressible flow equations using the Galerkin finite element method’, in et al. (eds), Finite Elements in Fluids, Vol. 4, Wiley, Chichester, 1982, pp. 21-45.
[16] ’Time integration and conjugate gradient methods for the incompressible Navier-Stokes equation’, Proc. 6th Int. Conf. on Finite Elements in Water Resources, Springer-Verlag, Lisbon, June 1986, pp. 3-29.
[17] Chorin, Math. Comput. 22 pp 745– (1968)
[18] Kim, J. Comput. Phys. 59 pp 308– (1985)
[19] and , ’Accurate explicit finite element schemes for convective-conductive heat transfer problem’, AMD Vol. 34, Finite Element Methods for Convection-Dominated Flows, ASME, 1979, pp. 149-166.
[20] Donea, Comput. Methods Appl. Mech. Eng. 30 pp 53– (1982)
[21] Gresho, Int. j. numer. methods Eng. 17 pp 790– (1982)
[22] Sani, Int. j. numer. methods fluids 1 pp 171– (1981)
[23] Temam, Arch. Rat. Mech. Anal. 32 pp 135– (1969)
[24] Ghia, J. Comput. Phys. 48 pp 387– (1982)
[25] Davis, J. Fluid Mech. 116 pp 475– (1982)
[26] Davis, Phys. Fluids 27 pp 46– (1984)
[27] Okajima, J. Fluid Mech. 123 pp 379– (1982)
[28] FIDAP Users Manual, Version 4.0, Fluid Dynamics International, Evanston, IL, 1987.
[29] Vortex Flow in Nature and Technology, Wiley, New York, 1983.
[30] Steger, AIAA J. 15 pp 581– (1977)
[31] Logan, AIAA J. 9 pp 660– (1971)
[32] Donea, J. Comput. Phys. 27 pp 463– (1987)
[33] Baker, Int. j. numer. methods fluids 7 pp 489– (1987)
[34] and , ’Incompressible flow computations based on the vorticity-stream function and velocity-pressure formulations’, UMSI 89/86, University of Minnesota, Supercomputer Institute, Minneapolis, MN, May 1989.
[35] Hughes, Int. j. numer. methods fluids 7 pp 1261– (1987)
[36] Gresho, Int. j. numer. methods fluids 7 pp 1111– (1987)
[37] ’Incompressible fluid dynamics: some fundamental formulation issues’, in et al., (eds), Annual Review of Fluid Mechanics, Annual Reviews, Inc., Palo Alto, Calif., U.S.A., 1991. · Zbl 0717.76006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.