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The Laplace and the linear elasticity problems near polyhedral corners and associated eigenvalue problemsMeyer, Arnd, Pester, Cornelia 01 September 2006 (has links) (PDF)
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.
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Analytical investigations and numerical experiments for singularly perturbed convection-diffusion problems with layers and singularities using a newly developed FE-softwareLudwig, Lars 14 March 2014 (has links) (PDF)
In the field of singularly perturbed reaction- or convection-diffusion boundary value problems the research area of a priori error analysis for the finite element method, has already been thoroughly investigated. In particular, for mesh adapted methods and/or various stabilization techniques, works have been done that prove optimal rates of convergence or supercloseness uniformly in the perturbation parameter epsilon.
Commonly, however, it is assumed that the exact solution behaves nicely in that it obeys certain regularity assumptions although in general, e.g. due to corner singularities, these regularity requirements are not satisfied. So far, insufficient regularity has been met by assuming compatibility conditions on the data.
The present thesis originated from the question: What can be shown if these rather unrealistic additional assumptions are dropped? We are interested in epsilon-uniform a priori estimates for convergence and superconvergence that include some regularity parameter that is adjustable to the smoothness of the exact solution.
A major difficulty that occurs when seeking the numerical error decay is that the exact solution is not known. Since we strive for reliable rates of convergence we want to avoid the standard approach of the "double-mesh principle". Our choice is to use reference solutions as a substitute for the exact solution.
Numerical experiments are intended to confirm the theoretical results and to bring further insights into the interplay between layers and singularities. To computationally realize the thereby arising demanding practical aspects of the finite element method, a new software is developed that turns out to be particularly suited for the needs of the numerical analyst. Its design, features and implementation is described in detail in the second part of the thesis.
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The Laplace and the linear elasticity problems near polyhedral corners and associated eigenvalue problemsMeyer, Arnd, Pester, Cornelia 01 September 2006 (has links)
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.
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A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3D vertex singularitiesPester, Cornelia 07 May 2006 (has links) (PDF)
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.
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Analytical investigations and numerical experiments for singularly perturbed convection-diffusion problems with layers and singularities using a newly developed FE-softwareLudwig, Lars 04 March 2014 (has links)
In the field of singularly perturbed reaction- or convection-diffusion boundary value problems the research area of a priori error analysis for the finite element method, has already been thoroughly investigated. In particular, for mesh adapted methods and/or various stabilization techniques, works have been done that prove optimal rates of convergence or supercloseness uniformly in the perturbation parameter epsilon.
Commonly, however, it is assumed that the exact solution behaves nicely in that it obeys certain regularity assumptions although in general, e.g. due to corner singularities, these regularity requirements are not satisfied. So far, insufficient regularity has been met by assuming compatibility conditions on the data.
The present thesis originated from the question: What can be shown if these rather unrealistic additional assumptions are dropped? We are interested in epsilon-uniform a priori estimates for convergence and superconvergence that include some regularity parameter that is adjustable to the smoothness of the exact solution.
A major difficulty that occurs when seeking the numerical error decay is that the exact solution is not known. Since we strive for reliable rates of convergence we want to avoid the standard approach of the "double-mesh principle". Our choice is to use reference solutions as a substitute for the exact solution.
Numerical experiments are intended to confirm the theoretical results and to bring further insights into the interplay between layers and singularities. To computationally realize the thereby arising demanding practical aspects of the finite element method, a new software is developed that turns out to be particularly suited for the needs of the numerical analyst. Its design, features and implementation is described in detail in the second part of the thesis.
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A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3D vertex singularitiesPester, Cornelia 21 April 2006 (has links)
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.
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