The Finite Element Method for Hyperbolic Systems

  • O. Pironneau
Conference paper
Part of the ICASE/NASA LaRC Series book series (ICASE/NASA)


The finite element method (FEM) [32] was invented by structural analysts; to replace a continuous structure by many equivalent beams is a process that does not leave an engineer’s intuition at a loss.


Finite Element Method Hyperbolic System Quadrature Formula Advection Equation Artificial Viscosity 
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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  1. 1.
    F. Angrand, V. Billey, A. Dervieux, J. Periaux, C. Pouletty, and B. Stoufflet, “2-D and 3-D Duler flow calculations with a second-order accurate Galerkin finite element method,” AIAA18th Fluid Dynamics and Plasmadynamics and Lasers Conference (Cincinnati Ohio, July16 – 18, 1985 ).Google Scholar
  2. 2.
    K. Baba and T. Tabata, “on a conservative upwind finite element scheme for convective diffusion equations,” RAIRO Anal. Numér. 15, No. 1 (1981), 3 – 25MATHMathSciNetGoogle Scholar
  3. 3.
    T.J. Baker and A. Jameson, “A novel finite element method for the calculation of inviscid flow over a complete aircraft.” (R. Glowinski, ed.), Proceedings of the Conference on Finite Elements in Flow Problems (Nice, 1986 ).Google Scholar
  4. 4.
    Y. Brenier and G. Cohen, “Transport of contour lines with mixed finite elements for 2 phase flows,” in Numerical Methods for Transient Coupled Problems(K.W. Lewis, ed. ), Pineridge Press, 1984, pp. 741 – 755.Google Scholar
  5. 5.
    M.O. Bristeau, R. Glowinski, J. Periaux, P. Perrier, and O. Pironneau, “On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods,” in Proceedings of FENOMECH’78, Comp. Meth. Appl. Mech. Eng., 17/18 (1979), part 3.“ Formulations for convection dominated flows with particular emphasis on the incompressible Navier—Stokes equations,” Comput. Methods Appl Mech. Engrg., 32(1982), 199 – 259.Google Scholar
  6. 6.
    A.N. Brooks and T. Hughes, “Streamline upwind/Petrov—Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier—Stokes equations.” Comput. Methods in Appl. Mech. Engrg., 32(1982), 199 – 259.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    G. Chavent and J. JafTre, “Mathematical models and finite elements for reservoir simulation,” Stud. Math. Appl. (1986), to appear.Google Scholar
  8. 8.
    I. Christie, D.F. Griffiths, A.R. Mitchell, and O.C. Zienckiewicz, “Finite element methods for second-order differential equations with significant first derivatives,” Internat. J. Numer. Methods Engrg., 10(1976), 1389 – 1396.CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    P. Ciarlet: The Finite Element Method, North-Holland, Amsterdam, 1979.Google Scholar
  10. 10.
    A. Dervieux, “Steady Euler simulations using unstructured meshes,” in Computational Fluid Dynamics, Von Karman Institute for Fluid Dynamics, Lecture Series 1985–04, March 26 – 29, 1985.Google Scholar
  11. 11.
    A. Dervieux, J.A. Desideri, F. Fezoui, B. Palmerio, and J.R. Rosenblum, “Euler calculations by upwind finite element methods,” GAMM Workshop, 1986, to appear. See also Numerical Methods for the Euler Equations of Fluid Dynamics(Dervieux et al, eds.) SIAM, Philadephia, 1985.Google Scholar
  12. 12.
    J. Douglas and T. Russell, “Numerical methods for convection dominated diffusion problems based on combining the methods of characteristics and the finite element methods,.” SIAM J. Numer. Anal., 19(1982), 871 – 885.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    M. Fortin and F. Thomasset, “Mixed finite element methods for incompressible flow problems,” J. Comput. Phys., 31, No. 1 (1979), 113 – 145.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    J.C. Heinrich, P.S. Huyakorn, O.C. Zienckiewicz, and A.R. Mitchell, “An upwind finite element scheme for the two-dimensional convective equation,” Internat. J. Numer. Methods Engrg., 11(1977), 131 – 143.CrossRefMATHGoogle Scholar
  15. 15.
    T.J.R. Hughes, “A simple finite element scheme for developping upwind finite elements,” Internat. J. Numer. Methods Engrg., 12(1978), 1359 – 1365.CrossRefMATHGoogle Scholar
  16. 16.
    T.J.R. Hughes, M. Mallet, and L.P. Franca, “Entropy-stable methods for compressible fluids: Application to high Mach number flows with shocks,” Finite Element Methods for Nonlinear Problems(Bergan, Bathe, Wunderlich, eds.), Springer-Verlag, Berlin, 1986.Google Scholar
  17. 17.
    P. Hood and C. Taylor, “A numerical solution of the Navier-Stokes equations using the finite element technique,” Comput. & Fluids, 1(1973), 73 – 100.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    C. Johnson, “Streamline diffusion methods for problems in fluid mechanics,” in Finite Elements in Fluids, Vol. 6, ( R.H. Gallagher, G.F. Carey, J.T. Oden, and O.C. Zienkiewicz, eds.), Wiley, New York, 1985.Google Scholar
  19. 19.
    C. Johnson and A. Szepessy, “A shock-capturing streamline diffusion finite element method for a nonlinear hyperbolic conservation law,” Dept. of Mathematics, Chalmers University of Technology and the University of Göteborg, S-412 96, Goteborg, Sweden, 1986.Google Scholar
  20. 20.
    C. Johnson and J. Pitkaranta, “An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation,” Math. Comput. 46, No. 173, (1986), 1 – 26.CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    P. Lesaint, “On a finite element for solving the neutron transport equation.” Math Aspects of FEM in PDE (C. de Boor, ed.), Academic Press, New York, 1974, pp. 89 – 123.Google Scholar
  22. 22.
    P. Lesaint, “Sur la resolution des systemes hyperboliques du premier ordre par des methodes d’èléments finis,” Thèse d’Etat, Université Pierre et Marie Curie, 1975.Google Scholar
  23. 23.
    P. Lesaint and P.A. Raviart, “Finite element collocation methods for first-order systems,” Math. Comp., 33, No. 147 (1979), 891 – 918.MATHMathSciNetGoogle Scholar
  24. 24.
    A. Mizukami and T.J.R. Hughes, “A Petrov—Galerkin finite element method for convection-dominated flows—an accurate upwinding technique for satisfying the maximum principle,” Comput. Methods Appl. Mech. Engrg., 50(1985), 181 – 193.CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    M. Mallet, “A finite element method for computational fluid dynamics,” Ph.D. Thesis, Standard, University 1985.Google Scholar
  26. 26.
    O. Pironneau, “On the transport diffusion algorithm and its applications to the Navier-Stokes equations,” Numer. Math., 38(1982), 309 – 332.CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    T. Russell, “An incomplete iterated characteristic finite element method for the miscible displacement problem.” Ph.D. Thesis, University of Chicago, 1980.Google Scholar
  28. 28.
    A. Priestley, “Lagrange and characteristic Galerkin methods for evolution problems,” D.Phil. Thesis, Oxford University 1986.Google Scholar
  29. 29.
    F. Thomasset, Implementation of Finite Element Methods for the Navier—Stokes Equations, Springer Lecture Notes in Computational Physics, 1981.Google Scholar
  30. B. Van LEER, Computational methods for ideal compressible flows, VKI Lecture Series, 04, 1983.Google Scholar
  31. 31.
    S.F. Wornom and M.M. Hafez, “Calculation of quasi-one-dimensional flows with shocks,” Comput. & Fluids, 14, No. 2 (1986), 131 – 140.CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    O.C. Zienkiewicz, The Finite Element Method in Engineering Science, McGraw- Hill, New York, 1977.Google Scholar
  33. 33.
    O.C. Zienkiewicz and J.C. Heinrich, “Quadratic finite element scheme for two- dimensional convective transport problem,” Internat. J. Numer. Methods Engrg., 11(1977), 1831 – 1844.CrossRefMATHGoogle Scholar
  34. 34.
    O.C. Zienkiewicz, K. Morgan, J. Peraire, M. Vahdati, and R. Lohner, “Finite elements for compressible gas flow and similar systems,” Proceedings of the 1th International Conference on Computational Methods, (Versailles, 1985 ). ( R. Glowinski, ed.), North-Holland, Amsterdam 1986.Google Scholar
  35. 35.
    R. Löhner, K. Morgan, and O. Zienkiewicz, “The solution of nonlinear systems of hyperbolic equations by the finite element method,” Internat. J. Numer. Methods Fluids 4(1984), 1043 – 1063.CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    F. Fezoui, B. Stoufflet, J. Periaux, and A. Dervieux, “Implicit high-order upwind finite element schemes for the Euler equations.” GAMNI, Conference (Atlanta, 1986 ).Google Scholar
  37. 37.
    J.T. Oden, S.J. Robertson, T. Strouboulis, P. Devloo, L.W. Spradey, and H.V. Mc Connaughey, “Adaptive and moving mesh finite element methods for flow interaction problems,” 6th International Symposium on Finite Element in Flow Problems (Antibe, 1986) (M.O. Briteau et al eds.),.Google Scholar
  38. 38.
    K.W. Morton and B.W. Scotney, “Petrov—Galerkin methods and diffusion— convection problems in 2D,” The Math, of FEM and Appl V, Proc. MAFELAP 1984, (Whiteman, ed.), Academic Press, New York, 1985, pp. 343 – 366.Google Scholar
  39. 39.
    K.W. Morton, Finite element methods for non-self-adjoint problems, Lecture Notes in Mathematics, Vol. 965 (P.R. Turner, ed.), Springer-Verlag, New York, 1982, pp. 113 – 148.Google Scholar
  40. 40.
    B. Palmerio, V. Billey, A. Dervieux, and J. Periaux “Self-adaptive mesh refinements and FEM for solving the Euler equation,” in Numerical Methods for Fluid Dynamics, Vol. II (K. Morton ed.), Clarendon Press, Oxford, 1986, pp. 369 – 388.Google Scholar
  41. 41.
    Scotney, “Petrov—Galerkin methods and diffusion—convection problems in 2D,” The Mathematics of FEM and Appl. V Proc., MAFELAP 1984 (Whiteman, ed.), Academic Press, New York, 1985, pp. 343 – 366.Google Scholar

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© Springer Science+Business Media New York  1988

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  • O. Pironneau

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