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A Survey of Mixed Finite Element Methods

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Part of the book series: ICASE/NASA LaRC Series ((ICASE/NASA))

Abstract

The aim of this paper is to present an introductory survey of mixed finite element methods. We shall deal first with the so-called mixed formulation of some problems arising in elasticity and hydrodynamics. Then we shall analyze the difficulties connected with the choice of appropriate finite element discretizations for a mixed formulation. Finally, we shall discuss some special techniques that are often helpful for solving the discretized problem.

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References

  1. M. Amara and J.M. Thomas, “Equilibrium finite elements for the linear elastic problem,” Numer. Math., 33(1979), 367 – 383.

    Article  MATH  MathSciNet  Google Scholar 

  2. D.N. Arnold and F. Brezzi, “Mixed and nonconforming finite element methods: Implementation, postprocessing, and error estimates,” RAIRO M 2 AN, 19(1985), 7 – 32.

    MATH  MathSciNet  Google Scholar 

  3. D.N. Arnold, F. Brezzi, and J. Douglas, Jr., “PEERS: A new mixed finite element for plane elasticity,” Japan J. Appl. Math., 1(1984), 347 – 367.

    Article  MATH  MathSciNet  Google Scholar 

  4. D.N. Arnold, J. Douglas, Jr., and C.P. Gupta, “A family of higher order finite element methods for plane elasticity,” Numer. Math. 45(1984), 1 – 22.

    Article  MATH  MathSciNet  Google Scholar 

  5. D.N. Arnold and R.S. Falk, “A new mixed formulation for elasticity,” Numer. Math., to appear.

    Google Scholar 

  6. I. Babuška, “The finite element method with Lagrangian multipliers,” Numer. Math., 20(1973), 179 – 192.

    Article  MATH  Google Scholar 

  7. I. Babuška and J.E. Osborn, “Numerical treatment of eigenvalue problems for differential equations with discontinuous coefficients,” Math. Comp., 32(1978), 991 – 1023.

    MATH  MathSciNet  Google Scholar 

  8. M. Bercovier, These de Doctorat d’Etat, Université de Rouen, 1976.

    Google Scholar 

  9. C. Bernardi, C. Canuto, and Y. Maday, “Generalized inf—sup condition for Chebyshev approximation of the Navier—Stokes equations,” submitted to SIAM J. Numer. Anal.

    Google Scholar 

  10. J. Boland and R. Nicolaides, “Stability of finite elements under divergence constraints,” SIAM J. Numer. Anal, 20, No. 4 (1983), 722 – 731.

    Article  MATH  MathSciNet  Google Scholar 

  11. F. Brezzi, “On the existence, uniqueness, and approximation of saddle-point problems arising from Lagrangian multipliers,” RAIRO Anal Numér., 8–32(1974), 129 – 151.

    MathSciNet  Google Scholar 

  12. F. Brezzi and J. Douglas, Jr., “Stabilized mixed methods for the Stokes problem,” Numer. Math., to appear.

    Google Scholar 

  13. F. Brezzi, J. Douglas, Jr., R. Duran, and M. Fortin, “Mixed finite elements for second-order elliptic problems in three variables,” Numer. Math., 51(1987), 237 – 250.

    Article  MATH  MathSciNet  Google Scholar 

  14. F. Brezzi, J. Douglas, Jr., M. Fortin, and L.D. Marini, “Efficient rectangular mixed finite elements in two and three space variables,” RAIRO M 2AN, 21, No. 4 (1987), 581 – 604.

    MATH  MathSciNet  Google Scholar 

  15. F. Brezzi, J. Douglas, Jr., and L.D. Marini, “Two families of mixed finite elements for second-order elliptic problems,” Numer. Math., 47(1985), 217 – 235.

    Article  MATH  MathSciNet  Google Scholar 

  16. F. Brezzi and J. Pitkaranta, “On the stabilization of finite element approximations of the Stokes equations,” Efficient Solutions of Elliptic Systems(Kiel, 1984 ), Braunschweig, Vieweg, 1984.

    Google Scholar 

  17. P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.

    MATH  Google Scholar 

  18. M. Crouzeix and P.A. Raviart, “Conforming and non-conforming finite element methods for solving the stationary Stokes equations,” RAIRO Anal Numér., R3(1973) 33 – 76.

    MathSciNet  Google Scholar 

  19. J. Douglas, Jr. and J. E. Roberts, “Mixed finite element methods for second-order elliptic problems,” Mat. Apl. Comput., 1(1982), 91 – 103.

    MATH  MathSciNet  Google Scholar 

  20. J. Douglas, Jr. and J. E. Roberts, “Global estimates for mixed methods for second- order elliptic equations,” Math. Comp., 44(1985), 39 – 52.

    MATH  MathSciNet  Google Scholar 

  21. R.S. Falk and J.E. Osborn, “Error estimates for mixed methods,” RAIRO Anal. Numér., 14(1980), 309 – 324.

    MathSciNet  Google Scholar 

  22. M. Fortin, “An analysis of the convergence of mixed finite element methods,” RAIRO Anal Numér., 11(1977), 341 – 354.

    MATH  MathSciNet  Google Scholar 

  23. B.X. Fraeijs de Veubeke, “Stress function approach,” in World Congress on the Finite Element Method in Structural Mechanics (Bournemouth, 1975.).

    Google Scholar 

  24. B.X. Fraeijs de Veubeke, “Displacement and equilibrium models in the finite element method,” in Stress Analysis( O.C. Zienkiewicz and G. Hollister, eds.), Wiley, New York, 1965.

    Google Scholar 

  25. L.P. Franca, T.J.R. Hughes, A.F.D. Loula, and I. Miranda, “A new family of stable elements for nearly incompressible elasticity: A mixed Petrov-Galerkin finite element method,” Numér. Math., to appear.

    Google Scholar 

  26. L. Gastaldi and R.H. Nochetto, “Optimal L∞-error estimates for nonconforming and mixed finite element methods of lowest order,” Numer. Math., 50(1987), 587 – 611.

    Article  MATH  MathSciNet  Google Scholar 

  27. L. Gastaldi and R.H. Nochetto, “On L8 -accuracy of mixed finite element methods for second-order elliptic problems,” Mat. Apl. Comput., to appear.

    Google Scholar 

  28. V. Girault and P. A. Raviart, “Finite element approximation of the Navier—Stokes equations,” Lecture Notes in Mathematics, Vol. 749, Springer-Verlag, New York, 1979.

    Google Scholar 

  29. R. Glowinski, “Approximations externes par elements finis d’ordre un et deux du probleme de Dirichlet pour Δ2,” in Topics in Numerical Analysis, Vol. I (J.J.H. Miller, ed.), Academic Press, London, 1973, pp. 123 – 171.

    Google Scholar 

  30. R. Glowinski and O. Pironneau, “Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem,” SIAM Rev., 21, No. 2 (1979), 167 – 212.

    Article  MATH  MathSciNet  Google Scholar 

  31. R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Comparative Physics, Springer-Verlag, New York, 1984.

    Book  Google Scholar 

  32. K. Hellan, “Analysis of elastic plates in flexure by a simplified finite element method,” Acta Poly tech. Scand. Math. Comput. Sci. Ser., 46(1967).

    Google Scholar 

  33. L.R. Herrmann, “Finite element bending analysis for plates,” J. Engrg. Mech. Div. ASCE EMS, 93(1967), 49 – 83.

    Google Scholar 

  34. L.R. Herrmann, “A bending analysis for plates,” in Proceedings of the Conference on Matrix Methods in Structural Mechanics, AFFDL-TR-66-88, pp. 577–604.

    Google Scholar 

  35. T.J.R. Hughes, L.P. Franca, M. Balestra, “A new finite element formulation for computational fluid dynamics, V: Circumventing the Babuska—Brezzi condition: A stable Petrov—Galerkin formulation of the Stokes problem accommodating equal order interpolations,” Comput. Meth. Appl Mech. Engrg., 59(1986), 85 – 101.

    Article  MATH  MathSciNet  Google Scholar 

  36. C. Johnson, “On the convergence of some mixed finite element methods for plate bending problems,” Numer. Math., 21(1973), 43 – 62.

    Article  MATH  MathSciNet  Google Scholar 

  37. C. Johnson and B. Mercier, “Some equilibrium finite element methods for two- dimensional elasticity problems,” Numer. Math., 30(1978), 103 – 116.

    Article  MATH  MathSciNet  Google Scholar 

  38. C. Johnson and J. Pitkaranta, “Analysis of some mixed finite element methods related to reduced integration,” Math. Comp., 38(1982), 375 – 400.

    Article  MATH  MathSciNet  Google Scholar 

  39. L.D. Marini, “An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method,” SIAM J. Numer. Anal, 22, No. 3 (1985), 493 – 496.

    Article  MATH  MathSciNet  Google Scholar 

  40. L.D. Marini and A. Savini, “Accurate computation of electric field in reverse- biased semiconductor devices: A mixed finite element approach,” COMPEL, 3, No. 3 (1984), 123 – 135.

    Article  MATH  Google Scholar 

  41. J.C. Nedelec, “Mixed finite elements in ℝ3,” Numer. Math., 35(1980), 315 – 341.

    Article  MATH  MathSciNet  Google Scholar 

  42. J.C. Nedelec, “A new family of mixed finite elements in ℝ3,” Numer. Math., 50(1986), 57 – 82.

    Article  MATH  MathSciNet  Google Scholar 

  43. J. Pitkaranta and R. Stenberg, “Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes,” in MAFELAP V (Brunei, 1984, J. Whiteman, ed.), Academic Press, London, 1985, pp. 325 – 334.

    Google Scholar 

  44. P.A. Raviart and J.M. Thomas, “A mixed finite element method for second-order elliptic problems,” in Mathematical Aspects of Finite Element Methods(Rome, 1975), Lecture Notes in Mathematics, Vol. 606, Springer-Verlag, New York, 1975, pp. 292 – 315.

    Google Scholar 

  45. R. Scholz, “L∞-convergence of saddle-point approximations for second-order problems,” RAIRO Anal. Numér., 11, No. 2 (1977), 209 – 216.

    MATH  MathSciNet  Google Scholar 

  46. R. Scholz, “Optimal L∞-estimates for a mixed finite element method for elliptic and parabolic problems,” Calcolo, 20(1983), 355 – 377.

    Article  MATH  MathSciNet  Google Scholar 

  47. L.R. Scott and M. Vogelius, “Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials,” RAIRO M 2 AN, 19(1985), 111 – 143.

    MATH  MathSciNet  Google Scholar 

  48. R. Stenberg, “Analysis of mixed finite element methods for the Stokes problem: A unified approach,” Math. Comp., 42(1984), 9 – 23.

    MATH  MathSciNet  Google Scholar 

  49. R. Verfürth, “Finite element approximation of stationary Navier—Stokes equa¬tions with slip boundary conditions,” Ruhr Universitatat Bochum, B. 75, June 1986

    Google Scholar 

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Brezzi, F. (1988). A Survey of Mixed Finite Element Methods. In: Dwoyer, D.L., Hussaini, M.Y., Voigt, R.G. (eds) Finite Elements. ICASE/NASA LaRC Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3786-0_2

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  • DOI: https://doi.org/10.1007/978-1-4612-3786-0_2

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-8350-8

  • Online ISBN: 978-1-4612-3786-0

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