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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 (UGent)
Author
Organization
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.
Keywords
EIGENVALUE PROBLEM, MULTICOMPONENT DOMAIN, OPERATOR METHOD, NUMERICAL INTEGRATION, FINITE ELEMENT APPROXIMATION

Citation

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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.
@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},
}