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

DOI:
https://doi.org/10.1090/S0025-5718-1986-0842120-4

MathSciNet review:
842120

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**J. T. Beale & A. Majda, "Vortex methods. I and II,"*Math. Comp.*, v. 39, 1982, pp. 1-27 and 29-52. MR**658213 (83i:65069b)****[2]**P. G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam, 1978. MR**0520174 (58:25001)****[3]**K. Eriksson, C. Johnson & V. Thomée, "Time discretization of parabolic problems by the discontinuous Galerkin method,"*Math. Modelling and Numer. Anal.*, v. 19, 1985, pp. 611-643. MR**826227 (87e:65073)****[4]**V. Girault & P. A. Raviart, "An analysis of a mixed finite element method for the Navier-Stokes equations,"*Numer. Math.*, v. 33, 1979, pp. 235-271. MR**553589 (81a:65100)****[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]**C. Johnson & U. Nävert, "An analysis of some finite element methods for advection-diffusion," in*Analytical and Numerical Approaches to Asymptotic Problems in Analysis*(O. Axelsson, L. S. Frank and A. Van der Sluis, eds.), North-Holland, Amsterdam, 1981. MR**605494 (81m:65005)****[9]**C. Johnson, "Finite element methods for convection-diffusion problems," in*Computing Methods in Engineering and Applied Sciences*. V (R. Glowinski and J. L. Lions, eds.), North-Holland, Amsterdam, 1981. MR**784648 (86d:65124)****[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 & J. Pitkäranta, "An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation,"*Math. Comp.*, v. 46, 1986, pp. 1-26. MR**815828 (88b:65109)****[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. Lesaint & P. A. Raviart, "On a finite element method for solving the neutron transport equation," in*Mathematical Aspects of Finite Elements in Partial Differential Equations*(C. de Boor, ed.), Academic Press, New York, 1974, pp. 89-123. MR**0658142 (58:31918)****[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 solutions of the Euler equations of incompressible perfect fluids," in*Turbulence and Navier Stokes Equations*(R. Temam, ed.), Lecture Notes in Math., Vol. 565, Springer, Berlin and New York, 1976. MR**0467033 (57:6902)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30,
76-08,
76D05

Retrieve articles in all journals with MSC: 65N30, 76-08, 76D05

Additional Information

DOI:
https://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