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A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations. (English) Zbl 0716.76048

Summary: A streamline diffusion finite element method is introduced for the time- dependent incompressible Navier-Stokes equations in a bounded domain in \({\mathbb{R}}^ 2\) and \({\mathbb{R}}^ 3\) in the case of high Reynolds number flow. An error estimate is proved and numerical results are given. The method is based on a mixed velocity-pressure formulation using the same finite element discretization of space-time for the velocity and the pressure spaces, which consist of piecewise linear functions, together with certain least-squares modifications of the Galerkin variational formulation giving added stability without sacrificing accuracy.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76M30 Variational methods applied to problems in fluid mechanics
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[1] Brooks, A.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 32, 199-259 (1982) · Zbl 0497.76041
[2] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg., 45, 285-312 (1984) · Zbl 0526.76087
[3] P. Hansbo, Adaptivity and streamline diffusion procedures in the finite element method, Thesis, Publication 89:2, Department of Structural Mechanics, Chalmers University of Technology, Sweden.; P. Hansbo, Adaptivity and streamline diffusion procedures in the finite element method, Thesis, Publication 89:2, Department of Structural Mechanics, Chalmers University of Technology, Sweden.
[4] Hughes, T. J.R.; Mallet, M.; Mizukami, A., A new finite element formulation for computational fluid dynamics: II. Beyond SUPG, Comput. Methods Appl. Mech. Engrg., 54, 341-355 (1986) · Zbl 0622.76074
[5] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. The general streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 58, 305-328 (1986) · Zbl 0622.76075
[6] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 58, 329-336 (1986) · Zbl 0587.76120
[7] Hughes, T. J.R.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg., 73, 173-189 (1989) · Zbl 0697.76100
[8] Johnson, C.; Saranen, J., Streamline diffusion methods for problems in fluid mechanics, Math. Comp., 47, 1-18 (1986)
[9] Johnson, C.; Szepessy, A.; Hansbo, P., On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp., 54, 107-130 (January 1990)
[10] Johnson, C.; Szepessy, A., On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp., 49, 427-444 (1987) · Zbl 0634.65075
[11] Szepessy, A., Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions, Math. Comp., 53, 527-545 (October 1989)
[12] Szepessy, A., Measure valued solutions of scalar conservation laws with boundary conditions, Arch. Rational Mech. Anal., 107, 2, 181-193 (1989) · Zbl 0702.35155
[13] A. Szepessy, Convergence of the streamline diffusion finite element method for conservation laws. Thesis, Department of Mathematics, Chalmers University of Technology. Sweden.; A. Szepessy, Convergence of the streamline diffusion finite element method for conservation laws. Thesis, Department of Mathematics, Chalmers University of Technology. Sweden. · Zbl 0751.65061
[14] P. Hansbo, Finite element procedures for conduction and convection problems, Publication 86:7, Department of Structural Mechanics, Chalmers University of Technology, Sweden.; P. Hansbo, Finite element procedures for conduction and convection problems, Publication 86:7, Department of Structural Mechanics, Chalmers University of Technology, Sweden.
[15] Bercovier, M.; Pironneau, O., Error estimates for finite element method solution of the Stokes problem in the primitive variables, Numer. Math., 33, 211-224 (1979) · Zbl 0423.65058
[16] Brezzi, F.; Pitkäranta, J., On the stabilization of finite element approximations of the Stokes equations, (Hackbusch, W., Efficient Solutions of Elliptic Systems. Efficient Solutions of Elliptic Systems, Notes on Numerical Fluid Mechanics, Vol. 10 (1984), Vieweg: Vieweg Braunschweig), 11-19 · Zbl 0552.76002
[17] Girault, V.; Raviart, P.-A., Finite Element Approximation of the Navier-Stokes Equations, (Lecture Notes in Mathematics, Vol. 749 (1979), Springer: Springer Berlin) · Zbl 0413.65081
[18] Pitkäranta, J.; Stenberg, R., Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes, (Whiteman, J. R., The Mathematics of Finite Elements and Applications. The Mathematics of Finite Elements and Applications, V. MAFELAP 1984 (1985), Academic Press: Academic Press New York), 325-334 · Zbl 0598.76036
[19] Stenberg, R., Analysis of mixed finite element methods for the Stokes problem: A unified approach, Math. Comp., 42, 9-23 (1984) · Zbl 0535.76037
[20] Franca, L. P.; Hughes, T. J.R., Two classes of mixed finite element methods, Comput. Methods Appl. Mech. Engrg., 69, 89-129 (1988) · Zbl 0651.65078
[21] Hughes, T. J.R.; Franca, L. P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg., 59, 85-99 (1986) · Zbl 0622.76077
[22] Hughes, T. J.R.; Franca, L. P., A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces, Comput. Methods Appl. Mech. Engrg., 65, 85-96 (1987) · Zbl 0635.76067
[23] Brezzi, F.; Douglas, J., Stabilized mixed methods for the Stokes problem, Numer. Math., 53, 225-235 (1988) · Zbl 0669.76052
[24] Fortin, M.; Glowinski, R., Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems (1983), North-Holland: North-Holland Amsterdam · Zbl 0525.65045
[25] Soh, W. Y.; Goodrich, J. W., Unsteady solution of incompressible Navier-Stokes equations, J. Comput. Phys., 79, 113-143 (1988) · Zbl 0651.76012
[26] Tezduyar, T. E.; Liou, J.; Ganjoo, D. K., Incompressible flow computations based on the vorticity-stream function and velocity-pressure formulations, Minnesota Supercomputer Institute Report UMSI 89/86 (May 1989)
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