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Streamline diffusion methods for the incompressible Euler and Navier-Stokes equations


Authors: Claes Johnson and Jukka Saranen
Journal: Math. Comp. 47 (1986), 1-18
MSC: Primary 65N30; Secondary 76-08, 76D05
MathSciNet review: 842120
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Abstract: We present and analyze extensions of the streamline diffusion finite element method to the time-dependent two-dimensional Navier-Stokes equations for an incompressible fluid in the case of high Reynolds numbers. The limit case with zero viscosity, the Euler equations, is also considered.


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  • [1] J. Thomas Beale and Andrew Majda, Vortex methods. II. Higher order accuracy in two and three dimensions, Math. Comp. 39 (1982), no. 159, 29–52. MR 658213, 10.1090/S0025-5718-1982-0658213-7
  • [2] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • [3] Kenneth Eriksson, Claes Johnson, and Vidar Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 4, 611–643 (English, with French summary). MR 826227
  • [4] V. Girault and P.-A. Raviart, An analysis of a mixed finite element method for the Navier-Stokes equations, Numer. Math. 33 (1979), no. 3, 235–271. MR 553589, 10.1007/BF01398643
  • [5] T. J. Hughes & A. Brooks, "A multidimensional upwind scheme with no crosswind diffusion," in AMD, v. 34, Finite Element Methods for Convection Dominated Flows (T. J. Hughes, ed.), ASME, New York, 1979.
  • [6] T. J. Hughes & A. Brooks, "A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: Application to the streamline-upwind procedure," Finite Elements in Fluids, Vol. 4 (R. H. Gallagher, ed.), Wiley, New York, 1982.
  • [7] T. J. Hughes, E. T. Tezduyar & A. Brooks, Streamline Upwind Formulation for Advection-Diffusion, Navier-Stokes and First Order Hyperbolic Equations, Fourth Internat. Conf. on Finite Element Methods in Fluids, Tokyo, July, 1982.
  • [8] O. Axelsson, L. S. Frank, and A. van der Sluis (eds.), Analytical and numerical approaches to asymptotic problems in analysis, North-Holland Mathematics Studies, vol. 47, North-Holland Publishing Co., Amsterdam-New York, 1981. MR 605494
  • [9] Claes Johnson, Finite element methods for convection-diffusion problems, Computing methods in applied sciences and engineering, V (Versailles, 1981), North-Holland, Amsterdam, 1982, pp. 311–323. MR 784648
  • [10] C. Johnson, U. Nävert & J. Pitkäranta, "Finite element methods for linear hyperbolic problems," Comput. Methods Appl. Mech. Engrg., v. 45, 1985, pp. 285-312.
  • [11] C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), no. 173, 1–26. MR 815828, 10.1090/S0025-5718-1986-0815828-4
  • [12] C. Johnson, Error Estimates and Automatic Time Step Control for Numerical Methods for Stiff Ordinary Differential Equations, Technical report, Chalmers Univ. of Technology, Göteborg, 1984.
  • [13] P. Lesaint, Sur la Résolution des Systèmes Hyperboliques du Premier Ordre par des Méthodes d'Élements Finis, Thèse, Université Paris VI, 1975.
  • [14] P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. Publication No. 33. MR 0658142
  • [15] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.
  • [16] U. Nävert, A Finite Element for Convection-Diffusion Problems, Thesis, Chalmers Univ. of Technology, Göteborg, 1982.
  • [17] R. Temam, Local existence of 𝐶^{∞} solutions of the Euler equations of incompressible perfect fluids, Turbulence and Navier-Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975) Springer, Berlin, 1976, pp. 184–194. Lecture Notes in Math., Vol. 565. MR 0467033

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1986-0842120-4
Keywords: Finite element method, incompressible flow, time-dependent, high Reynolds number
Article copyright: © Copyright 1986 American Mathematical Society