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An operator method for a numerical quadrature finite element approximation for a class of second-order elliptic eigenvalue problems in composite structures

Michèle Vanmaele UGent and Roger Van Keer (1995) ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE. 29(3). p.339-365
abstract
We consider a second-order elliptic eigenvalue problem on a convex polygonal domain, divided in M non-overlapping subdomains. The conormal derivative of the unknown function is continuous on the interfaces, while the function itself is discontinuous. In this paper we study the finite element approximation without and with numerical quadrature of this eigenvalue problem by means of the perturbation theory for linear compact, self-adjoint operptors, see [13, IV.2-IV.3, V.4.3] and [9]. We refine the method, developed in [18], by incorporating some basic ideas of [4] and [9]. This improved method is then extended to the underlying multi-component structure with discontinuities at the interfaces. Furthermore, in contrast to [3] and [4], which are dealing with a Dirichlet eigenvalue problem on a one-component domain, discretized by a triangular mesh, we allow for a rectangular mesh, for mired Dirichlet-Robin boundary conditions and for a more general second-order differential operator Finally, in contrast to [18], we also consider finite elements of higher degree (quadratic, biquadratic,...). Crucial to the finite element analysis is a non-standard variational formulation to the eigenvalue problem, similar to the one in [II] for some classes of parabolic problems. The emphasis of this paper is on the error analysis of the approximate eigenpairs.
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author
organization
alternative title
An operator method for a numerical quadrature finite element approximation for a class of 2nd-order elliptic eigenvalue problems in composite structures
year
type
journalArticle (original)
publication status
published
subject
keyword
EIGENVALUE PROBLEM, MULTICOMPONENT DOMAIN, OPERATOR METHOD, NUMERICAL INTEGRATION, FINITE ELEMENT APPROXIMATION
journal title
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE
ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer.
volume
29
issue
3
pages
339 - 365
Web of Science type
Article
ISSN
0764-583X
language
English
UGent publication?
yes
classification
A1
id
194537
handle
http://hdl.handle.net/1854/LU-194537
date created
2004-01-14 13:42:00
date last changed
2017-09-28 12:04:58
@article{194537,
  abstract     = {We consider a second-order elliptic eigenvalue problem on a convex polygonal domain, divided in M non-overlapping subdomains. The conormal derivative of the unknown function is continuous on the interfaces, while the function itself is discontinuous. In this paper we study the finite element approximation without and with numerical quadrature of this eigenvalue problem by means of the perturbation theory for linear compact, self-adjoint operptors, see [13, IV.2-IV.3, V.4.3] and [9]. We refine the method, developed in [18], by incorporating some basic ideas of [4] and [9]. This improved method is then extended to the underlying multi-component structure with discontinuities at the interfaces. Furthermore, in contrast to [3] and [4], which are dealing with a Dirichlet eigenvalue problem on a one-component domain, discretized by a triangular mesh, we allow for a rectangular mesh, for mired Dirichlet-Robin boundary conditions and for a more general second-order differential operator Finally, in contrast to [18], we also consider finite elements of higher degree (quadratic, biquadratic,...). 
Crucial to the finite element analysis is a non-standard variational formulation to the eigenvalue problem, similar to the one in [II] for some classes of parabolic problems. The emphasis of this paper is on the error analysis of the approximate eigenpairs.},
  author       = {Vanmaele, Mich{\`e}le and Van Keer, Roger},
  issn         = {0764-583X},
  journal      = {ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE},
  keyword      = {EIGENVALUE PROBLEM,MULTICOMPONENT DOMAIN,OPERATOR METHOD,NUMERICAL INTEGRATION,FINITE ELEMENT APPROXIMATION},
  language     = {eng},
  number       = {3},
  pages        = {339--365},
  title        = {An operator method for a numerical quadrature finite element approximation for a class of second-order elliptic eigenvalue problems in composite structures},
  volume       = {29},
  year         = {1995},
}

Chicago
Vanmaele, Michèle, and Roger Van Keer. 1995. “An Operator Method for a Numerical Quadrature Finite Element Approximation for a Class of Second-order Elliptic Eigenvalue Problems in Composite Structures.” Esaim-mathematical Modelling and Numerical Analysis-modelisation Mathematique Et Analyse Numerique 29 (3): 339–365.
APA
Vanmaele, Michèle, & Van Keer, R. (1995). An operator method for a numerical quadrature finite element approximation for a class of second-order elliptic eigenvalue problems in composite structures. ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE, 29(3), 339–365.
Vancouver
1.
Vanmaele M, Van Keer R. An operator method for a numerical quadrature finite element approximation for a class of second-order elliptic eigenvalue problems in composite structures. ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE. 1995;29(3):339–65.
MLA
Vanmaele, Michèle, and Roger Van Keer. “An Operator Method for a Numerical Quadrature Finite Element Approximation for a Class of Second-order Elliptic Eigenvalue Problems in Composite Structures.” ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE 29.3 (1995): 339–365. Print.