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, Volume 36 (2002) no. 6, p. 1043-1070
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.
DOI : https://doi.org/10.1051/m2an:2003005
Classification:  65N25,  65N30,  74G70
Keywords: quadratic eigenvalue problems, linear elasticity, 3D vertex singularities, finite element methods, error estimates
@article{M2AN_2002__36_6_1043_0,
     author = {Apel, Thomas and S\"andig, Anna-Margarete and Solov'ev, Sergey I.},
     title = {Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {6},
     year = {2002},
     pages = {1043-1070},
     doi = {10.1051/m2an:2003005},
     zbl = {1137.65426},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2002__36_6_1043_0}
}
Apel, Thomas; Sändig, Anna-Margarete; 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, Volume 36 (2002) no. 6, pp. 1043-1070. doi : 10.1051/m2an:2003005. http://www.numdam.org/item/M2AN_2002__36_6_1043_0/

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