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The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited

Y. Latushkin, A. Sukhtayev (2010)

Mathematical Modelling of Natural Phenomena

This paper is related to the spectral stability of traveling wave solutions of partial differential equations. In the first part of the paper we use the Gohberg-Rouche Theorem to prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstract operator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of the corresponding Birman-Schwinger type operator pencil. In the second part of the paper we apply this result...

The BC-method in Multidimensional Spectral Inverse Problem: Theory and Numerical Illustrations

M. I. Belishev, V. Yu. Gotlib, S. A. Ivanov (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This work is devoted to numerical experiments for multidimensional Spectral Inverse Problems. We check the efficiency of the algorithm based on the BC-method, which exploits relations between Boundary Control Theory and Inverse Problems. As a test, the problem for an ellipse is considered. This case is of interest due to the fact that a field of normal geodesics loses regularity on a nontrivial separation set. The main result is that the BC-algorithm works quite successfully in spite of...

The change in electric potential due to lightning

William W. Hager, Beyza Caliskan Aslan (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

The change in the electric potential due to lightning is evaluated. The potential along the lightning channel is a constant which is the projection of the pre-flash potential along a piecewise harmonic eigenfunction which is constant along the lightning channel. The change in the potential outside the lightning channel is a harmonic function whose boundary conditions are expressed in terms of the pre-flash potential and the post-flash potential along the lightning channel. The expression for the...

The density of states of a local almost periodic operator in ℝ ν

Andrzej Krupa (2003)

Studia Mathematica

We prove the existence of the density of states of a local, self-adjoint operator determined by a coercive, almost periodic quadratic form on H m ( ℝ ν ) . The support of the density coincides with the spectrum of the operator in L ² ( ℝ ν ) .

The effect of reduced integration in the Steklov eigenvalue problem

Maria G. Armentano (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we analyze the effect of introducing a numerical integration in the piecewise linear finite element approximation of the Steklov eigenvalue problem. We obtain optimal order error estimates for the eigenfunctions when this numerical integration is used and we prove that, for singular eigenfunctions, the eigenvalues obtained using this reduced integration are better approximations than those obtained using exact integration when the mesh size is small enough.

The effect of reduced integration in the Steklov eigenvalue problem

María G. Armentano (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we analyze the effect of introducing a numerical integration in the piecewise linear finite element approximation of the Steklov eigenvalue problem. We obtain optimal order error estimates for the eigenfunctions when this numerical integration is used and we prove that, for singular eigenfunctions, the eigenvalues obtained using this reduced integration are better approximations than those obtained using exact integration when the mesh size is small enough.

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