Convergence and Error Estimates for a Finite Element Method with Numerical Quadrature for a Second Order Elliptic Eigenvalue Problem

  • Michèle Vanmaele
  • Roger Van Keer
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique book series (ISNM, volume 96)


This paper deals with a FE-numerical quadrature method for a class of 2nd order elliptic eigenvalue problems on a bounded rectangular domain Ω ⊂ ℝ2, viz.
$$ {\text{Find }}\lambda {\text{ }} \in {\text{ }}\mathbb{R}{\text{, u }} \in {\text{ V : a(u,v) = }}\lambda {\text{. (u,v) }}\forall {\text{ v }} \in {\text{ V}} $$
$$ \begin{array}{*{20}{c}} {{\text{V}}\{ {\text{v}} \in {\text{ }}{{{\text{H}}}^{1}}{\text{(}}\Omega ){\text{|v = 0on}}{{\Gamma }_{1}}{\text{ }} \subset {\text{ }}\partial \Omega {\text{ = boundary of }}\Omega \} } \\ {{\text{(}}{{\Gamma }_{1}}{\text{ consisting of an integer number of sides)}}} \\ {(.,.) = {{{\text{L}}}_{{\text{2}}}}{\text{ (}}\Omega {\text{) - inner product}}} \\ {{\text{a(u,v) = }}\int\limits_{\Omega } {[\sum\nolimits_{{{\text{i,j = l}}}}^{2} {{{{\text{a}}}_{{{\text{ij}}}}}{\text{(x)}}\cdot \frac{{\partial {\text{u}}}}{{\partial {{{\text{x}}}_{{\text{i}}}}}}\cdot \frac{{\partial {\text{v}}}}{{\partial {{{\text{x}}}_{{\text{j}}}}}}{\text{ + }}{{{\text{a}}}_{0}}{\text{(x)}}{\text{.u}}.{\text{v}}]{\text{.dx, x = (}}{{{\text{x}}}_{1}}{\text{,}}{{{\text{x}}}_{2}}{\text{)}}{\text{.}}} } } \\ \end{array} $$


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andreev, A.B., Kaschieva, V.A. and Vanmaele, M. (1990) Some results in lumped mass finite element approximation of eigenvalue problems using numerical quadrature, (submitted)Google Scholar
  2. 2.
    Babuska, I. and Osborn, J.E. (1989) Finite element-Galerkin approximation of the eigenvalues of selfadjoint problems. Math. Comp. 52, 275–297.Google Scholar
  3. 3.
    Bramble, J.H. and Hubert, S. (1971) Bounds for a class of linear functionals with applications to Hermite interpolation. Numer. Math. 16. 362–369.CrossRefGoogle Scholar
  4. 4.
    Ciariet, P.G. (1976) The finite element method for elliptic problems (North-Holland, Amsterdam).Google Scholar
  5. 5.
    Dautray, R. and Lions, J.L. (1985) Analyse mathématique et calcul numérique pour les sciences et les techniques, Tome 2 (Masson, Paris).Google Scholar
  6. 6.
    Davis, P.J. and Rabinowitz, P. (1975) Methods of numerical integration (Academic Press, N.Y.).Google Scholar
  7. 7.
    Fix, G.J. (1972) Effects of quadrature errors in finite element approximation of steady state, eigenvalue and parabolic problems. In: Aziz, A.K. (editor) The mathematical foundations of the finite element method with applications to partial differential equations (Academic Press, N.Y.).Google Scholar
  8. 8.
    Vanmaele M. and Van Keer R. (1990) On an internal approximation of a class of elliptic eigenvalue problems. In: Kurzweil, J. (editor) Equadiff 7 (Teubner, Leipzig) (to appear).Google Scholar

Copyright information

© Springer Basel AG 1991

Authors and Affiliations

  • Michèle Vanmaele
    • 1
  • Roger Van Keer
    • 1
  1. 1.Faculty of Engineering SciencesState University of GhentGentBelgium

Personalised recommendations