Finite element approximation of a contact vector eigenvalue problem

Hennie de Schepper; Roger Van Keer

Applications of Mathematics (2003)

  • Volume: 48, Issue: 6, page 559-571
  • ISSN: 0862-7940

Abstract

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We consider a nonstandard elliptic eigenvalue problem of second order on a two-component domain consisting of two intervals with a contact point. The interaction between the two domains is expressed through a coupling condition of nonlocal type, more specifically, in integral form. The problem under consideration is first stated in its variational form and next interpreted as a second-order differential eigenvalue problem. The aim is to set up a finite element method for this problem. The error analysis involved is shown to be affected by the nonlocal condition, which requires a suitable modification of the vector Lagrange interpolant on the overall finite element mesh. Nevertheless, we arrive at optimal error estimates. In the last section, an illustrative numerical example is given, which confirms the theoretical results.

How to cite

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de Schepper, Hennie, and Keer, Roger Van. "Finite element approximation of a contact vector eigenvalue problem." Applications of Mathematics 48.6 (2003): 559-571. <http://eudml.org/doc/33168>.

@article{deSchepper2003,
abstract = {We consider a nonstandard elliptic eigenvalue problem of second order on a two-component domain consisting of two intervals with a contact point. The interaction between the two domains is expressed through a coupling condition of nonlocal type, more specifically, in integral form. The problem under consideration is first stated in its variational form and next interpreted as a second-order differential eigenvalue problem. The aim is to set up a finite element method for this problem. The error analysis involved is shown to be affected by the nonlocal condition, which requires a suitable modification of the vector Lagrange interpolant on the overall finite element mesh. Nevertheless, we arrive at optimal error estimates. In the last section, an illustrative numerical example is given, which confirms the theoretical results.},
author = {de Schepper, Hennie, Keer, Roger Van},
journal = {Applications of Mathematics},
keywords = {eigenvalue problem; nonlocal coupling condition; finite elements; nonlocal coupling condition; finite elements},
language = {eng},
number = {6},
pages = {559-571},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Finite element approximation of a contact vector eigenvalue problem},
url = {http://eudml.org/doc/33168},
volume = {48},
year = {2003},
}

TY - JOUR
AU - de Schepper, Hennie
AU - Keer, Roger Van
TI - Finite element approximation of a contact vector eigenvalue problem
JO - Applications of Mathematics
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 6
SP - 559
EP - 571
AB - We consider a nonstandard elliptic eigenvalue problem of second order on a two-component domain consisting of two intervals with a contact point. The interaction between the two domains is expressed through a coupling condition of nonlocal type, more specifically, in integral form. The problem under consideration is first stated in its variational form and next interpreted as a second-order differential eigenvalue problem. The aim is to set up a finite element method for this problem. The error analysis involved is shown to be affected by the nonlocal condition, which requires a suitable modification of the vector Lagrange interpolant on the overall finite element mesh. Nevertheless, we arrive at optimal error estimates. In the last section, an illustrative numerical example is given, which confirms the theoretical results.
LA - eng
KW - eigenvalue problem; nonlocal coupling condition; finite elements; nonlocal coupling condition; finite elements
UR - http://eudml.org/doc/33168
ER -

References

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  1. 10.1016/0362-546X(93)90183-S, Nonlinear Anal. 20 (1993), 27–61. (1993) MR1199063DOI10.1016/0362-546X(93)90183-S
  2. Introduction à l’analyse numérique des équations aux dérivées partielles (2ième tirage), Masson, Paris, 1993. (1993) 
  3. The Finite Element Method for Elliptic Problems, North Holland Publishing Company, Amsterdam, 1978. (1978) Zbl0383.65058MR0520174
  4. An operator method for a numerical quadrature finite element approximation for a class of second-order elliptic eigenvalue problems in composite structures, RAIRO Model. Math. Anal. Numer. 29 (1995), 339–365. (1995) MR1342711

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