Advertisement

Elements of Functional and Numerical Analysis

  • Dominique Chapelle
  • Klaus-Jürgen Bathe
Chapter
Part of the Computational Fluid and Solid Mechanics book series (COMPFLUID)

Abstract

A deeper understanding of finite element methods, and the development of improved finite element methods, can only be achieved with an appropriate mathematical and numerical assessment of the proposed techniques. The basis of such an assessment rests on identifying whether certain properties are satisfied by the finite element scheme and these properties depend on the framework within which the finite element method has been formulated.

In this chapter we first review fundamental concepts of functional analysis, and then present different basic frameworks of variational formulations and finite element discretizations that we will use in the later chapters for shell solutions. For completeness, we prove the stability and convergence properties of the abstract finite element discretizations for each of the frameworks of variational formulations considered. This chapter therefore provides the foundation used for the later assessment of the reliability and effectiveness of shell finite element schemes.

Keywords

Sobolev Space Bilinear Form Variational Formulation Mixed Formulation Finite Element Approximation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Chapelle, D., & Bathe, K.J. (1993). The inf-sup test. Comput. & Structures, 47(4/5), 537-545.CrossRefzbMATHMathSciNetGoogle Scholar
  2. Lions, J.L., & Magenes, E. (1972). Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Berlin: Springer-Verlag.Google Scholar
  3. Lax, P.D., & Milgram, A.N. (1954). Parabolic equations. In Annals of Mathematics Studies, 33, (pp. 167-190). Princeton: Princeton University Press.Google Scholar
  4. Brenner, S.C., & Scott, L.R. (1994). The Mathematical Theory of Finite Element Methods. New York: Springer-Verlag.zbMATHGoogle Scholar
  5. Glowinski, R., & Le Tallec, P. (1989). Augmented Lagrangian and Operator- Splitting Methods in Nonlinear Mechanics. SIAM Studies in Applied Mathematics. Philadelphia: SIAM.zbMATHGoogle Scholar
  6. Ainsworth, M., & Oden, J.T. (2000). A Posteriori Error Estimation in Finite Element Analysis. New York: John Wiley & Sons.zbMATHGoogle Scholar
  7. Roberts, J.E., & Thomas, J.M. (1991). Mixed and hybrid methods. In P. Ciarlet, & J. Lions (Eds.) Handbook of Numerical Analysis, Vol. II . Amsterdam: North-Holland.Google Scholar
  8. Bathe, K.J. (1996). Finite Element Procedures. Englewood Cliffs: Prentice Hall.Google Scholar
  9. Adams, R.A. (1975). Sobolev Spaces. New York: Academic Press.zbMATHGoogle Scholar
  10. Banach, S. (1932). Théorie des Opérations Linéaires. Warszawa.Google Scholar
  11. Brezzi, F., & Fortin, M. (1991). Mixed and Hybrid Finite Element Methods. New York: Springer-Verlag.zbMATHGoogle Scholar
  12. Ciarlet, P.G., & Raviart, P.A. (1972). The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. In A. Aziz (Ed.) The Mathematical Foundations of the Finite Element Method with Applications to Partial Diérential Equations, (pp. 409-474). New York: Academic Press.Google Scholar
  13. Arnold, D.N., & Brezzi, F. (1993). Some new elements for the Reissner-Mindlin plate model. In J. Lions, & C. Baiocchi (Eds.) Boundary Value Problems for Partial Diérential Equations and Applications, (pp. 287-292). Paris: Masson.Google Scholar
  14. Ciarlet, P.G. (1978). The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland.zbMATHGoogle Scholar
  15. Lee, N.S., & Bathe, K.J. (1993). Eécts of element distortions on the performance of isoparametric elements. Internat. J. Numer. Methods Engrg., 36, 3553-3576.CrossRefzbMATHGoogle Scholar
  16. Dautray, R., & Lions, J.L. (1988-1993). Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1-6. Berlin: Springer-Verlag.Google Scholar
  17. Chapelle, D., & Bathe, K.J. (2010). On the ellipticity condition for modelparameter dependent mixed formulations. Comput. & Structures, 88, 581-587. Doi:10.1016/j.compstruc.2010.01.009.CrossRefGoogle Scholar
  18. Bathe, K.J., Bucalem, M.L., & Brezzi, F. (1990). Displacement and stress convergence of our MITC plate bending elements. Eng. Comput., 7, 291-302.CrossRefGoogle Scholar
  19. Brezzi, F. (1974). On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers. R.A.I.R.O., Anal. Numér., 8, 129-151.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dominique Chapelle
    • 1
  • Klaus-Jürgen Bathe
    • 2
  1. 1.INRIA Paris-RocquencourtLe ChesnayFrance
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations