Numerical Approximation of PDEs and Clément’s Interpolation

  • Jacques Rappaz
Part of the Operator Theory: Advances and Applications book series (OT, volume 168)


In this short paper, we present a formalism which specifies the notions of consistency and stability of finite element methods for the numerical approximation of nonlinear partial differential equations of elliptic and parabolic type. This formalism can be found in [4], [7], [10], and allows to establish a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods. In concrete cases, the Cléement’s interpolation technique [6] is very useful in order to establish local a posteriori error estimates. This paper uses some ideas of [10] and its main goal is to show in a very simple setting, the mathematical arguments which lead to the stability and convergence of Galerkin methods. The bibliography concerning this subject is very large and the references of this paper are no exhaustive character. In order to obtain a large bibliography on the a posteriori error estimates, we report the lecturer to Verfürth’s book and its bibliography [12].


Numerical Approximation Posteriori Error Posteriori Error Estimate Nonsmooth Function Local Regularization 
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  1. [1]
    I. Babuska, A.K. Aziz. Survey lectures on the mathematical foundations of the finite element method. Academic Press, New York and London (1972).Google Scholar
  2. [2]
    C. Bernardi. Optimal finite element interpolation on curved domains. SIAM J. Numer. Anal. 26, 1212–1240 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    C. Bernardi, V. Girault. A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal., Vol.35, no 5, 1893–1916 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    G. Caloz, J. Rappaz. Numerical analysis for nonlinear and bifurcation problems. Handbook of Numerical Analysis, Vol. 5, P.G. Ciarlet and J.L. Lions ed., 487–637 (1997).Google Scholar
  5. [5]
    P.G. Ciarlet. Basic error estimates for elliptic problems. Handbook of numerical analysis, Vol.II, ed. P.G. Ciarlet and J.L. Lions, Elsevier, 17–351 (1991).Google Scholar
  6. [6]
    P. Clément. Approximation by finite element functions using local regularization. RAIRO Anal. Num. 2, 77–84 (1975).Google Scholar
  7. [7]
    M. Crouzeix, J. Rappaz. On numerical approximation in bifurcation theory. Masson, Paris (1990).Google Scholar
  8. [8]
    G. Kunert. An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86, 471–490 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Medina, M. Picasso, J. Rappaz. Error estimates and adaptive finite elements for nonlinear diffusion-convection problems. M3AS 6-5, 689–712 (1996).Google Scholar
  10. [10]
    J. Pousin, J. Rappaz. Consistency, stability, a priori and a posteriori errors for Petrov-Galerkin methods applied to nonlinear problems. Numerische Mathematik 69, 213–231 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R. Scott, S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. of Comp., Vol. 54, no 190, 483–493 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    R. Verfürth. A review of a posteriori error estimation and adaptive mesh — refinement techniques. Wiley-Teubner series. Advances in numerical mathematics (1996).Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Jacques Rappaz
    • 1
  1. 1.Institute of Analysis and Scientific ComputingEcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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