Streamline diffusion methods for the incompressible Euler and NavierStokes equations
Authors:
Claes Johnson and Jukka Saranen
Journal:
Math. Comp. 47 (1986), 118
MSC:
Primary 65N30; Secondary 7608, 76D05
MathSciNet review:
842120
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Abstract: We present and analyze extensions of the streamline diffusion finite element method to the timedependent twodimensional NavierStokes 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.
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
(83i:65069b), http://dx.doi.org/10.1090/S00255718198206582137
 [2]
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1978. Studies in
Mathematics and its Applications, Vol. 4. MR 0520174
(58 #25001)
 [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
(87e:65073)
 [4]
V.
Girault and P.A.
Raviart, An analysis of a mixed finite element method for the
NavierStokes equations, Numer. Math. 33 (1979),
no. 3, 235–271. MR 553589
(81a:65100), http://dx.doi.org/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 PetrovGalerkin methods with discontinuous weighting functions: Application to the streamlineupwind 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 AdvectionDiffusion, NavierStokes 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, NorthHolland Mathematics Studies,
vol. 47, NorthHolland Publishing Co., AmsterdamNew York, 1981. MR 605494
(81m:65005)
 [9]
Claes
Johnson, Finite element methods for convectiondiffusion
problems, Computing methods in applied sciences and engineering, V
(Versailles, 1981), NorthHolland, Amsterdam, 1982,
pp. 311–323. 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. 285312.
 [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
(88b:65109), http://dx.doi.org/10.1090/S00255718198608158284
 [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 WisconsinMadison,
Academic Press, New York, 1974, pp. 89–123. Publication No. 33.
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 ConvectionDiffusion 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
NavierStokes equations (Proc. Conf., Univ. ParisSud, Orsay, 1975)
Springer, Berlin, 1976, pp. 184–194. Lecture Notes in Math.,
Vol. 565. MR
0467033 (57 #6902)
 [1]
 J. T. Beale & A. Majda, "Vortex methods. I and II," Math. Comp., v. 39, 1982, pp. 127 and 2952. MR 658213 (83i:65069b)
 [2]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, NorthHolland, 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. 611643. MR 826227 (87e:65073)
 [4]
 V. Girault & P. A. Raviart, "An analysis of a mixed finite element method for the NavierStokes equations," Numer. Math., v. 33, 1979, pp. 235271. 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 PetrovGalerkin methods with discontinuous weighting functions: Application to the streamlineupwind 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 AdvectionDiffusion, NavierStokes 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 advectiondiffusion," in Analytical and Numerical Approaches to Asymptotic Problems in Analysis (O. Axelsson, L. S. Frank and A. Van der Sluis, eds.), NorthHolland, Amsterdam, 1981. MR 605494 (81m:65005)
 [9]
 C. Johnson, "Finite element methods for convectiondiffusion problems," in Computing Methods in Engineering and Applied Sciences. V (R. Glowinski and J. L. Lions, eds.), NorthHolland, 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. 285312.
 [11]
 C. Johnson & J. Pitkäranta, "An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation," Math. Comp., v. 46, 1986, pp. 126. 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. 89123. 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 ConvectionDiffusion 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)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608421204
PII:
S 00255718(1986)08421204
Keywords:
Finite element method,
incompressible flow,
timedependent,
high Reynolds number
Article copyright:
© Copyright 1986
American Mathematical Society
