Singularity exponents

Singularity exponents for the Fichera corner

Singularity exponents for the Fichera corner

THE DOMAIN

The Fichera corner is the cube (-1,1) x (-1,1) x (-1,1) minus the cube [0,1] x [0,1] x [0,1]; that is 7/8 of a cube. This cube has one concave corner and three re-entrant edges which are of interest in the analysis of corner and edge singularities.

THE NAME

The Fichera corner is the prototype of a domain where edge and corner singularities interact. It is named after Gaetano Fichera (1922-1996) who was the first to give an approximation of its first corner singularity exponent: In 1973 he provided the bounds 0.4335 and 0.4645. Later, in 1993, the value 0.45418 was obtained by a boundary element method approximation.
(Source: http://perso.univ-rennes1.fr/monique.dauge/3eJS.html, http://www.icp.ac.ru/RAS_1724-1999/CD_PAH/ENG/48/4859.HTM)

SINGULARITY EXPONENTS

The singular parts of the solutions to boundary value problems in domains with concave corners are determined by so-called singularity exponents. These exponents are the eigenvalues of a (usually) quadratic operator eigenvalue problem which is define on the sphere of the three-dimensional unit ball.

We use the finite element method to discretize such eigenvalue problems. Because of the special spectral structure of these problems, efficient algorithms allow a fast computation of the singularity exponents. Some numerical results for the Fichera corner are summarized below. Further results can be found in the documentation of the program package CoCoS.

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NUMERICAL RESULTS

With the p-version of the finite element method and the SHIRA (Skew-Hamiltonian implicitly restarted Arnoldi) process, we obtained the following singularity exponents for the linear elasticity problem (isotropic material with the Poisson ratio 0.3) with Neumann boundary conditions:
0.76167112546691
0.76167113268264
0.77253377400496
These are preceded by the triple exponent 0.00 and followed by the triple exponent 1.00.
We cannot guarantee the validity of the last digits, but we assume these values are correct, at least, up to five or six digits.

The 3 smallest singularity exponents for the linear elasticity problem with Dirichlet boundary conditions are about
0.40438246700462
0.40442125126817
0.57411535942684

The 4 smallest singular exponents for the Laplace problem with Dirichlet boundary conditions were computed by CoCoS to be
0.45417371533061
1.23087025699075
1.23087029254928
1.78426613632728

The implementation of the finite element mesh in CoCoS allows considerations of variable opening angles (greater or less than 90 degree).


The computation of corner singularities is part of the project
Numerische Lösung von quadratischen Operator-Eigenwertproblemen aus der Kontinuumsmechanik.
We thank the DFG (Deutsche Forschungsgemeinschaft) for supporting this project.


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