# Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes

• Volume: 36, Issue: 6, page 1043-1070
• ISSN: 0764-583X

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## Abstract

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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.

## How to cite

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Apel, Thomas, Sändig, Anna-Margarete, and Solov'ev, Sergey I.. "Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.6 (2002): 1043-1070. <http://eudml.org/doc/245618>.

@article{Apel2002,
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.},
author = {Apel, Thomas, Sändig, Anna-Margarete, Solov'ev, Sergey I.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {quadratic eigenvalue problems; linear elasticity; 3D vertex singularities; finite element methods; error estimates},
language = {eng},
number = {6},
pages = {1043-1070},
publisher = {EDP-Sciences},
title = {Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes},
url = {http://eudml.org/doc/245618},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Apel, Thomas
AU - Sändig, Anna-Margarete
AU - Solov'ev, Sergey I.
TI - Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 6
SP - 1043
EP - 1070
AB - 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.
LA - eng
KW - quadratic eigenvalue problems; linear elasticity; 3D vertex singularities; finite element methods; error estimates
UR - http://eudml.org/doc/245618
ER -

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