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Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element methodon graded meshes

Published online by Cambridge University Press:  15 January 2003

Thomas Apel ,
Anna-Margarete Sändig  and
Sergey I. Solov'ev
Thomas Apel
Affiliation:
TU Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany. apel@mathematik.tu-chemnitz.de.
Anna-Margarete Sändig
Affiliation:
Universität Stuttgart, Mathematisches Institut A, Pfaffenwaldring 57, 70511 Stuttgart, Germany. saendig@mathematik.uni-stuttgart.de.
Sergey I. Solov'ev
Affiliation:
Kazan State University, Faculty of computer science and cybernetics, Kremlevskaya 18, 420008 Kazan, Russia. sergei.solovyev@ksu.ru.
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Abstract

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.

Keywords

Quadratic eigenvalue problemslinear elasticity3D vertex singularitiesfinite element methodserror estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 6 , November 2002 , pp. 1043 - 1070
Copyright
© EDP Sciences, SMAI, 2002

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