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.
Mots-clés : 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: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {1043--1070}, publisher = {EDP-Sciences}, volume = {36}, number = {6}, year = {2002}, doi = {10.1051/m2an:2003005}, zbl = {1137.65426}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2003005/} }
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: Modélisation mathématique et analyse numérique PY - 2002 SP - 1043 EP - 1070 VL - 36 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003005/ DO - 10.1051/m2an:2003005 LA - en ID - M2AN_2002__36_6_1043_0 ER -
%0 Journal Article %A Apel, Thomas %A Sändig, Anna-Margarete %A Solov'ev, Sergey I. %T Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes %J ESAIM: Modélisation mathématique et analyse numérique %D 2002 %P 1043-1070 %V 36 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003005/ %R 10.1051/m2an:2003005 %G en %F 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: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 6, pp. 1043-1070. doi : 10.1051/m2an:2003005. http://www.numdam.org/articles/10.1051/m2an:2003005/
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