zbMATH — the first resource for mathematics

Tuning the mesh of a mixed method for the stream function. Vorticity formulation of the Navier-Stokes equations. (English) Zbl 0760.76041
Summary: We study a new mixed method for the Stokes and Navier-Stokes equations. The method uses two meshes, one very fine for \(\omega\) and a coarser one for \(\psi\). Error estimates show that boundary layers do not require to refine the mesh for the stream function \(\psi\) as much as for the vorticity \(\omega\) when the Reynolds number is large. We prove estimates and study implementation problems.

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76D07 Stokes and related (Oseen, etc.) flows
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
Full Text: DOI EuDML
[1] Achdou, Y., Glowinski, R., Pironneau, O. (1991): Tunning the mesh of a mixed method for the ??? formulation of the Navier-Stokes equations. INRIA Report 1514 · Zbl 0760.76041
[2] Achdou, Y., Pironneau, O. (1991): The ?-method for the Navier-Stokes equations. C.R. Acad. Sci., Paris, Sér. I. 168 · Zbl 0736.76012
[3] Benqué, J.P., Haugel, A., Viollet, P.L. (1982): Engineering Applications of Hydrolics, Vol. II. Pitman, Montreal
[4] Bernardi, C., Girault, V., Maday, Y. (1991): Mixed Spectral Approximation of the Navier-Stokes equations. Université Paris 6, Report R 90045 · Zbl 0753.76130
[5] Bernardi, C., Godlewski, E., Raugel, G. (1987): A mixed method for the time dependant Navier-Stokes problem. IMA J. Numer. Anal.7, 165-189 · Zbl 0652.76018 · doi:10.1093/imanum/7.2.165
[6] Brezzi, F., Canuto, C., Russo, A. (1989): A self-adaptive formulation for the Euler/Navier-Stokes coupling. Comput. Methods Appl. Mech. Eng.73, 317-330 · Zbl 0688.76024 · doi:10.1016/0045-7825(89)90071-6
[7] Ciarlet, P.G., Lions, J.L. (1991): Handbook of numerical analysis, Vol. II. Elsevier/North Holland, Amsterdam New York · Zbl 0712.65091
[8] Ciarlet, P.G., Raviart P.A. (1974): A mixed method for the biharmonic equation. In: C. de Boor, ed., Mathematical aspect of finite elements in partial differential equations. Academic Press, New York, pp. 125-145 · Zbl 0337.65058
[9] Dean, E.J., Glowinski, R., Pironneau, O. (1991): Iterative solution of the stream function-vorticity formulation of the Stokes problem. Applications to the numerical simulation of incompressible viscous flow. Comput. Methods Appl. Mech. Eng.87, 117-155 · Zbl 0760.76044 · doi:10.1016/0045-7825(91)90003-O
[10] Douglas, J., Russell, T.F. (1982): Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element methods or finite difference method. SIAM J. Numer. Anal.19, 871-885 · Zbl 0492.65051 · doi:10.1137/0719063
[11] Fix, G.J., Strang, G. (1973): An analysis of the finite element method. Prentice-Hall, Englewood Cliffs, NJ · Zbl 0356.65096
[12] Girault, V., Raviart, P.A. (1986): Finite element methods for Navier-Stokes equations. Springer, Berlin Heidelberg New York · Zbl 0585.65077
[13] Glowinski, R., Pironneau, O. (1979): Numerical methods for the first biharmonic equation and for the two dimensional Stokes problem. SIAM Review121, 167-212 · Zbl 0427.65073 · doi:10.1137/1021028
[14] Landau, L., Lifchitz, F. (1971): Mécanique des fluides. MIR, Moscow
[15] Necas, J. (1967): Les méthodes directes en théorie des équations elliptiques. Masson, Paris
[16] Pironneau, O. (1989): Finite element for fluids, Wiley, Chichester · Zbl 0665.73059
[17] Pironneau, O., Rappaz, J. (1989): Numerical analysis for compressible viscous isentropic stationary flows. Impact1, pp. 109-137 · Zbl 0702.76073 · doi:10.1016/0899-8248(89)90026-8
[18] Rannacher, R., Scott, R. (1982): Some optimal error estimates for piecewise linear finite element approximations. Math. Comput.38, 435-445 · Zbl 0483.65007
[19] Rosenhead, L. (ed.) (1963): Laminar boundary layers. Oxford University Press, Oxford · Zbl 0115.20705
[20] Scholz, R. (1978): A mixed method for fourth order problem using finite elements. R.A.I.R.O. Anal. Numer.12, 85-90 · Zbl 0382.65059
[21] Suli, E. (1988): Convergence and non-linear stability of the Lagrange Galerkin method for the Navier-Stokes equations. Numer. Math.53, 459-483 · Zbl 0637.76024 · doi:10.1007/BF01396329
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.