Approximation in eigenvalue problems for holomorphic fredholm operator functions I

Karma, Otto

doi:10.1080/01630569608816699

Cited by 59 publications

(118 citation statements)

References 7 publications

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“…In order to estimate the last term, we need to calculate W n . If we replace (4.17) by 19) thenW n = δ(iκ n − s 0 )F iκ n due to (4.10). A matrix representation of both problems can be formulated using the matrices M long inf and S long inf of (3.16), whereas (4.17) leads to the N×N sub matrix of the N 0 ×N 0 matrix representation of (4.19).…”

Section: Improved Convergence Rates For Scattering Problemsmentioning

confidence: 99%

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Two scale Hardy space infinite elements for scalar waveguide problems

Halla

Nannen

2017

Adv Comput Math

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We consider the numerical solution of the Helmholtz equation in domains with one infinite cylindrical waveguide. Such problems exhibit wavenumbers on different scales in the vicinity of cut-off frequencies. This leads to performance issues for non-modal methods like the perfectly matched layer or the Hardy space infinite element method. To improve the latter, we propose a two scale Hardy space infinite element method which can be optimized for wavenumbers on two different scales. It is a tensor product Galerkin method and fits into existing analysis. Up to arbitrary small thresholds it converges exponentially with respect to the number of longitudinal unknowns in the waveguide. Numerical experiments support the theoretical error bounds.

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Section: Improved Convergence Rates For Scattering Problemsmentioning

confidence: 99%

“…The latter work, which is based on [19,20], derives a condition to ensure convergence of Galerkin methods to weakly T -coercive source and eigenvalue problems. The obtained condition is the existence of bounded linear operators T h,N on and to V h,N with lim…”

Section: Convergence Analysismentioning

confidence: 99%

Two scale Hardy space infinite elements for scalar waveguide problems

Halla

Nannen

2017

Adv Comput Math

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“…The approximation of Fredholm operator functions has been studied intensively [35,36] and the general theory provide basic convergence results. However, we will study the block operator matrix formulation (6).…”

Section: Galerkin Discretizationmentioning

confidence: 99%

“…For an overview of applications leading to nonlinear eigenvalue problems from a computational linear algebra point of view we refer to [10,50]. Approximations of operator functions were also extensively studied; and the general operator theory provides basic a-priori convergence results for Fredholm-valued functions [35,36,53].…”

Section: Introductionmentioning

confidence: 99%

Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications

Engström¹,

Giani

Grubišić

2016

Computers & Mathematics with Applications

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Citation for published item:ingstr¤ omD gF nd qiniD F nd qrui § si¡ D vF @PHITA 9i0ient nd relile hpEpiw estimtes for qudrti eigenvlue prolems nd photoni rystl pplitionsF9D gomputers nd mthemtis with pplitionsFD UP @RAF ppF WSPEWUQF Further information on publisher's website: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractWe present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one.

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“…The work for a Priori error estimates in [71] follows [41] and [42]. The basic idea is to construct the equivalent eigenvalue problems M for H and M n for P n H respectively.…”

Section: A Priori Error Estimatesmentioning

confidence: 99%

Boundary element approximation for Maxwell's eigenvalue problem

Wieners

Xin

2013

Math Methods in App Sciences

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We introduce a new method for computing eigenvalues of the Maxwell operator with boundary finite elements. On bounded domains with piecewise constant material coefficients, the Maxwell solution for fixed wave number can be represented by boundary integrals, which allows to reduce the eigenvalue problem to a nonlinear problem for determining the wave number along with boundary and interface traces. A Galerkin discretization yields a smooth nonlinear matrix eigenvalue problem that is solved by Newton's method or, alternatively, the contour integral method. Several numerical results including an application to the band structure computation of a photonic crystal illustrate the efficiency of this approach. Copyright © 2013 John Wiley & Sons, Ltd.

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Approximation in eigenvalue problems for holomorphic fredholm operator functions I

Cited by 59 publications

References 7 publications

Two scale Hardy space infinite elements for scalar waveguide problems

Two scale Hardy space infinite elements for scalar waveguide problems

Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications

Boundary element approximation for Maxwell's eigenvalue problem

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