The Laplace and the linear elasticity problems near polyhedral corners and associated eigenvalue problems

The solutions to certain elliptic boundary value problems have singularities with a typical structure near polyhedral corners. This structure can be exploited to devise an eigenvalue problem whose solution can be used to quantify the singularities of the given boundary value problem. It is necessary to parametrize a ball centered at the corner. There are different possibilities for a suitable parametrization; from the numerical point of view, spherical coordinates are not necessarily the best choice. This is why we do not specify a parametrization in this paper but present all results in a rather general form. We derive the eigenvalue problems that are associated with the Laplace and the linear elasticity problems and show interesting spectral properties. Finally, we discuss the necessity of widely accepted symmetry properties of the elasticity tensor. We show in an example that some of these properties are not only dispensable, but even invalid, although claimed in many standard books on linear elasticity

A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3D vertex singularities

This thesis is concerned with the finite element analysis and the a posteriori error estimation for eigenvalue problems for general operator pencils on two-dimensional manifolds. A specific application of the presented theory is the computation of corner singularities. Engineers use the knowledge of the so-called singularity exponents to predict the onset and the propagation of cracks. All results of this thesis are explained for two model problems, the Laplace and the linear elasticity problem, and verified by numerous numerical results

CoCoS - Computation of Corner Singularities

This is a documentation of the software package COCOS. The purpose of COCOS is the computation of corner singularities of elliptic equations in polyhedral corners and crack tips. COCOS provides a self-contained library for the generation of structured 2D finite element meshes, including various routines for mesh manipulation, as well as several algorithms for the solution of quadratic eigenvalue problems with Hamiltonian structure. These and further features will be described in this documentation

A residual a posteriori error estimator for the eigenvalue problem for the Laplace-Beltrami operator

The Laplace-Beltrami operator corresponds to the Laplace operator on curved surfaces. In this paper, we consider an eigenvalue problem for the Laplace-Beltrami operator on subdomains of the unit sphere in R3. We develop a residual a posteriori error estimator for the eigenpairs and derive a reliable estimate for the eigenvalues. A global parametrization of the spherical domains and a carefully chosen finite element discretization allows us to use an approach similar to the one for the two-dimensional case. In order to assure results in the quality of those for plane domains, weighted norms and an adapted Clément-type interpolation operator have to be introduced

Hamiltonian eigenvalue symmetry for quadratic operator eigenvalue problems

When the eigenvalues of a given eigenvalue problem are symmetric with respect to the real and the imaginary axes, we speak about a Hamiltonian eigenvalue symmetry or a Hamiltonian structure of the spectrum. This property can be exploited for an efficient computation of the eigenvalues. For some elliptic boundary value problems it is known that the derived eigenvalue problems have this Hamiltonian symmetry. Without having a specific application in mind, we trace the question, under which assumptions the spectrum of a given quadratic eigenvalue problem possesses the Hamiltonian structure

Clément-type interpolation on spherical domains - interpolation error estimates and application to a posteriori error estimation

In this paper, a mixed boundary value problem for the Laplace-Beltrami operator is considered for spherical domains in $R^3$, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (\varphi, \theta), such that functions on the unit sphere are considered as functions in these coordinates. Careful investigation leads to the introduction of a proper finite element space corresponding to an isotropic triangulation of the underlying domain on the unit sphere. Error estimates are proven for a Clément-type interpolation operator, where appropriate, weighted norms are used. The estimates are applied to the deduction of a reliable and efficient residual error estimator for the Laplace-Beltrami operator

Clément-Type Interpolation on Spherical Domains -- Interpolation Error Estimates and Application to a Posteriori Error Estimates

In this paper, a mixed boundary value problem for the Laplace-Beltrami operator is considered for spherical domains in R³, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (φ, θ), such that functions on the unit sphere are considered as functions in these coordinates. Careful investigation leads to the introduction of a proper finite element space corresponding to an isotropic triangulation of the underlying domain on the unit sphere. Error estimates are proven for a Clément-type interpolation operator, where appropriate, weighted norms are used. The estimates are applied to the deduction of a reliable and efficient residual error estimator for the Laplace-Beltrami operator

Edge Singularities and Structure of the 3-D Williams Expansion

The elastic solution in a vicinity of a re-entrant wedge can be described by a Williams like expansion in terms of powers of the distance to a point on the edge. This expansion has a particular structure due to the invariance of the problem by translation parallel to the edge. We show here that some terms, so-called primary solutions, derive directly from solutions to the 2-D corner problem posed in the orthogonal cross section of the domain. The others, baptized shadow functions, derive of the primary solutions by integration along the axis parallel to the edge. This 3-D Williams expansion is shown to be equivalent to the edge expansion proposed by Costabel et al. [M. Costabel, M. Dauge, Z. Yosibash, A quasidual function method for extracting edge stress intensit