Nonlinear eigenvalue problems arise in a variety of science and engineering applications and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton's method, contour integration, and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.

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The nonlinear eigenvalue problem

This paper is concerned with the fictitious eigenfrequency problem of the boundary integral equation methods when solving exterior acoustic problems. A contour integral method is used to convert the nonlinear eigenproblems caused by the boundary element method into ordinary eigenproblems. Since both real and complex eigenvalues can be extracted by using the contour integral method, it enables us to investigate the fictitious eigenfrequency problem in a new way rather than comparing the accuracy of numerical solutions or the condition numbers of boundary element coefficient matrices. The interior and exterior acoustic fields of a sphere with both Dirichlet and Neumann boundary conditions are taken as numerical examples. The pulsating sphere example is studied and all fictitious eigenfrequencies corresponding to the related interior problem are observed. The reasons are given for the usual absence of many fictitious eigenfrequencies in the literature. Fictitious eigenfrequency phenomena of the Kirchhoff‐Helmholtz boundary integral equation, its normal derivative formulation and the Burton‐ Miller formulation are investigated through the eigenvalue analysis. The actual effect of the Burton‐ Miller formulation on fictitious eigenfrequencies is revealed and the optimal choice of the coupling parameter is confirmed.

https://www.elsevier.com/journals/engineering-analysis-with-boundary-elements/0955-7997?generatepdf=true

Engineering Analysis with Boundary Elements

This Letter sets a road map towards an experimental realization of strong coupling between free electrons and photons and analytically explores entanglement phenomena that emerge in this regime. The proposed model unifies the strong-coupling predictions with known electron-photon interactions. Additionally, this Letter predicts a non-Columbic entanglement between freely propagating electrons. Since strong coupling can map entanglements between photon pairs onto photon-electron pairs, it may harness electron beams for quantum communication, thus far exclusive to photons.

Entanglements of Electrons and Cavity Photons in the Strong-Coupling Regime.

The boundary element method (BEM) in the context of acoustics or Helmholtz problems is reviewed in this paper. The basis of the BEM is initially developed for Laplace’s equation. The boundary integral equation formulations for the standard interior and exterior acoustic problems are stated and the boundary element methods are derived through collocation. It is shown how interior modal analysis can be carried out via the boundary element method. Further extensions in the BEM in acoustics are also reviewed, including half-space problems and modelling the acoustic field surrounding thin screens. Current research in linking the boundary element method to other methods in order to solve coupled vibro-acoustic and aero-acoustic problems and methods for solving inverse problems via the BEM are surveyed. Applications of the BEM in each area of acoustics are referenced. The computational complexity of the problem is considered and methods for improving its general efficiency are reviewed. The significant maintenance issues of the standard exterior acoustic solution are considered, in particular the weighting parameter in combined formulations such as Burton and Miller’s equation. The commonality of the integral operators across formulations and hence the potential for development of a software library approach is emphasised.

The Boundary Element Method in Acoustics: A Survey

We present a biorthogonal approach for modeling the response of localized electromagnetic resonators using quasinormal modes, which represent the natural, dissipative eigenmodes of the system with complex frequencies. For many problems of interest in optics and nanophotonics, the quasinormal modes constitute a powerful modeling tool, and the biorthogonal approach provides a coherent, precise, and accessible derivation of the associated theory, enabling an illustrative connection between different modeling approaches that exist in the literature.

Modeling electromagnetic resonators using quasinormal modes

In this paper, a rigorous convergence and error analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem is presented. The formulation of the eigenvalue problem in terms of a boundary integral equation yields a nonlinear boundary integral operator eigenvalue problem. This nonlinear eigenvalue problem and its Galerkin approximation are analyzed in the framework of eigenvalue problems for holomorphic Fredholm operator-valued functions. The convergence of the approximation is shown and quasi-optimal error estimates are presented. Numerical experiments are given confirming the theoretical results.

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Convergence Analysis of a Galerkin Boundary Element Method for the Dirichlet Laplacian Eigenvalue Problem

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some appropriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.

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A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

We analyze the approximation of a vibro-acoustic eigenvalue problem for an elastic body which is submerged in a compressible inviscid fluid in $\mathbb{R}^3$. As a model, the time-harmonic elastodynamic and the Helmholtz equation are used and are coupled in a strong sense via the standard transmission conditions on the interface between the solid and the fluid. Our approach is based on a coupling of the field equations for the solid with boundary integral equations for the fluid. The coupled formulation of the eigenvalue problem leads to a nonlinear eigenvalue problem with respect to the eigenvalue parameter since the frequency occurs nonlinearly in the used boundary integral operators for the Helmholtz equation. The nonlinear eigenvalue problem and its Galerkin discretization are analyzed within the framework of eigenvalue problems for Fredholm operator-valued functions where convergence is shown and error estimates are given. For the numerical solution of the discretized nonlinear matrix eigenvalue prob...

/pdf/coupled-finite-and-boundary-element-methods-for-fluid-solid-1zu1n11jft.pdf

Coupled finite and boundary element methods for fluid-solid interaction eigenvalue problems

The convergence analysis of iterative methods for nonlinear eigenvalue problems is in the most cases restricted either to algebraically simple eigenvalues or to polynomial eigenvalue problems. In this paper we consider two classical methods for general holomorphic eigenvalue problems, namely the nonlinear generalized Rayleigh quotient iteration (NGRQI) and the augmented Newton method. The analysis of the convergence order of both methods is based on the representation of the eigenvalues as poles of the resolvent. This approach was already chosen for the analysis of the NGRQI by Langer in [19] for a more general setting where such a representation of the eigenvalues had to be assumed. The convergence orders of both methods depend on the order which an eigenvalue has as pole of the resolvent. Both methods exhibit a local quadratic convergence order for semi-simple eigenvalues. For defective eigenvalues in general only a local linear convergence is possible. In numerical experiments the theoretical results are confirmed.

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Convergence Orders of Iterative Methods for Nonlinear Eigenvalue Problems

In this paper, boundary integral formulations for a time-harmonic acoustic scattering-resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering-resonance problems split in general into two parts. One part consists of scattering-resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering-resonances, which allows in practical computations a reliable and simple identification of the scattering-resonances in particular for non-convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

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Gerhard Unger

Papers

Convergence Analysis of a Galerkin Boundary Element Method for the Dirichlet Laplacian Eigenvalue Problem

A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

Coupled finite and boundary element methods for fluid-solid interaction eigenvalue problems

Convergence Orders of Iterative Methods for Nonlinear Eigenvalue Problems

Combined boundary integral equations for acoustic scattering-resonance problems