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Streamline Diffusion Finite Element Methods for Incompressible and Compressible Fluid Flow

  • Claes Johnson
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 12)

Abstract

We give an overview of some recent theoretical and computational results for streamline diffusion finite element methods applied to the incompressible Navier-Stokes equations with small viscosity and to some nonlinear hyperbolic conservation laws modelling compressible flow.

Keywords

Finite Element Method Computational Fluid Dynamic Compressible Flow Entropy Solution Compressible Euler Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    R.J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. 82(1983) pp. 27–70.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    K. Eriksson and C. Johnson, An adaptive finite element method for linear elliptic problems, Technical report, Chalmers University of Technology, to appear in Math. Comp.Google Scholar
  3. 3.
    P. Hansbo, Finite element procedures for conduction and convection problems, Publication 86:7, Dept. of Structural Mechanics, Chalmers Univ. of Technology.Google Scholar
  4. 4.
    A. Harten, On the symmetric form of systems of conservation laws with entropy. Journal of Computational Physics 49 (1983) pp. 151–164.MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    T.J.R. Hughes and A. Brooks, A multidimensional upwind scheme with no crosswing diffusion, in AMD vol 34, Finite Element Methods for Convection Dominated Flows (T.J. Hughes ed.), ASME, New York 1979.Google Scholar
  6. 6.
    T.J.R. Hughes, E.t. Tezduyar and 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.Google Scholar
  7. 7.
    T.J.R. Hughes and T.E. Tezduyar, Finite Element Methods for First-Order Hyperbolic Systems with Particular Emphasis on the Compressible Euler Equations, Computer Methods in Applied Mechanics and Engineering, Vol. 45(1984), pp. 217–284.Google Scholar
  8. 8.
    T.J.R. Hughes, M. Mallet and L.P. Franca, Entropy-stable finite element methods for compressible fluids: Application to high mach number flows with shocks, to appear in Finite Element Methods for Nonlinear Problems (eds. P. Bergan et al.), Springer-Verlag.Google Scholar
  9. 9.
    T.J.R. Hughes, L.P. Franca and M. Mallet, A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes Equations and the second law of thermodynamics, to appear in Computer Methods in Applied Mechanics and Engineering.Google Scholar
  10. 10.
    T.J.R. Hughes, M. Mallet and A. Mizukami, A new element formulation for computational fluid dynamics: II. Beyond SUPG, to appear in Computer Methods in Applied Mechanics and Engineering.Google Scholar
  11. 11.
    T.J.R. Hughes and M. Mallet, A new finite element formulation for computational fluid dynamics:: III. The generalized streamline operator for multidimensional advective-diffusive systems, to appear in Computer Methods in Applied Mechanics and Engineering.Google Scholar
  12. 12.
    T.J.R. Hughes and M. Mallet, A new finite element formulation for computational fluid dynamics: IV. A discontinuity capturing operator for multidimensional advective-diffusive systems, to appear in Computer Methods in Applied Mechanics and Engineering.Google Scholar
  13. 13.
    C. Johnson and U. Nävert, An analysis of some finite element methods for advection-diffusion, in Analytical and Numerical Approaches to Asymptotic Problems in Analysis (eds. Axelsson et al), North-Holland, 1981.Google Scholar
  14. 14.
    C. Johnson, U. Nävert and J. Pitkäranta, Finite element methods for linear hyperbolic problems, Computer Methods in Appl. Mech. Eng. 45(1985), pp. 285–312.CrossRefGoogle Scholar
  15. 15.
    C. Johnson and J. Saranen, Streamline diffusion methods for the incompressible Euler and Navier-Stokes equation, Math. Comp. 47(1986), pp.1–18.MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    C. Johnson, Streamline diffusion methods for problems in fluid mechanics, in Finite Elements in Fluids, ed. Gallagher et al, Wiley 1985.Google Scholar
  17. 17.
    C. Johnson and A. Szepessy, Convergence of a finite element method for a nonlinear hyperbolic conservation law, Technical report, Mathematics Dept. Chalmers Univ. of Technology, Göteborg, to appear in Math. Comp.Google Scholar
  18. 18.
    C. Johnson and A. Szepessy, On the convergence of streamline diffusion finite element methods for hyperbolic conservation laws, Proc. ASME conference, Anaheim dec. 1986.Google Scholar
  19. 19.
    U. Nävert, A Finite Element Method for Convection-Diffusion Problems, Thesis, Chalmers University of Technology, Göteborg 1982.Google Scholar
  20. 20.
    A. Szepessy, A streamline diffusion finite element method for the incompressible Navier-Stokes equations in three dimensions, to appear.Google Scholar
  21. 21.
    E. Tadmor, Skew-selfadjoint forms for systems of conservation laws, Journal of Mathematical Analysis and Applications, Vol. 103(1984), pp. 428–442.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    L. Tartar, Compensated compactness and applications to partial differential equations, in Research notes in Mathematics, Nonlinear analysis and mechanics: Heriot-Watts Symposium, Vol. 4. ed. R.J. Knops, Pitman Press (1979).Google Scholar
  23. 23.
    K. Eriksson and C. Johnson, An adaptive finite element method for linear advection problems, to appear.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Claes Johnson
    • 1
  1. 1.Chalmers University of TechnologyGöteborgSweden

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