Hardy-Raum Methoden zur numerischen Lvon Streu- und Resonanzproblemen auf unbeschrGebieten
01 Jan 2008-
About: The article was published on 2008-01-01 and is currently open access. It has received 5 citations till now.
Citations
More filters
01 Jan 1986
TL;DR: The representation of acoustic and electromagnetic fields the special theory of relativity radiation resonators the theory of waveguides refraction surface waves scattering by smooth objects diffraction by edges transient waves as discussed by the authors.
Abstract: The representation of acoustic and electromagnetic fields the special theory of relativity radiation resonators the theory of waveguides refraction surface waves scattering by smooth objects diffraction by edges transient waves. Appendices: Bessel functions Legendre functions Mathieu functions parabolic cylinder functions spheroidal functions tensor calculus asymptotic evaluation of integrals.
23 citations
30 Nov 2009
TL;DR: In this article, the Perfectly-Matched -Layer method (PML) is used to restrict the simulation problem onto a bounded computational domain, and an adaptive PML method which exhibits a good convergence even for critical problems where standard PML implementations fail is proposed.
Abstract: Optical technologies are ubiquitously used in hi-tech devices. As a common feature of such devices one finds structures with dimensions in the order of the wavelength of the used light. To design and produce such devices, the wave nature of light must be taken into account. Accordingly, robust simulation tools are required which are based on rigorously solving Maxwell's equations, the governing equations of light propagation within macroscopic media. This thesis contributes to the modeling and the numerical computation of light scattering problems: Light scattering problems are typically posed on the entire space. The Perfectly-Matched -Layer method (PML) is widely used to restrict the simulation problem onto a bounded computational domain. We propose an adaptive PML method which exhibits a good convergence even for critical problems where standard PML implementations fail. Besides the computation of the near field, that is the electromagnetic field within the computational domain, it is of major interest to evaluate the electromagnetic field in the exterior domain and to compute the far field. So far, this was numerically only possible for simple geometries such as homogeneous exterior domains or layered media. To deal with more complicated devices, for example with waveguide inhomogeneities, we develop an evaluation formula based on the PML solution which allows for an exterior domain field evaluation in a half space above the device. Finally, we generalize the PML method to problems with multiply structured exterior domains. The term “multiply structured exterior domain” is defined in this thesis and means that the exterior domain exhibits several half-infinite structures. Mathematically, this gives rise to various complications. For example, no analytical solutions to Maxwell's equations for standard light sources are available in the exterior domain, which are needed to describe the incoming field in a light scattering problem. To tackle this we propose a new light scattering problem formulation which fits well into the PML method framework and which may be regarded as an extension of classical contributions by Sommerfeld, Wiener and Hopf. An exterior domain evaluation formula for multiply structured exterior domains with an extended illumination is derived as well.
21 citations
Posted Content•
TL;DR: In this paper, a class of approximate transparent boundary conditions for the solution of Helmholtz-type resonance and scattering problems on unbounded domains is introduced, where the computational domain is assumed to be a polygon.
Abstract: This paper introduces a class of approximate transparent boundary conditions for the solution of Helmholtz-type resonance and scattering problems on unbounded domains. The computational domain is assumed to be a polygon. A detailed description of two variants of the Hardy space infinite element method which relays on the pole condition is given. The method can treat waveguide-type inhomogeneities in the domain with non-compact support. The results of the Hardy space infinite element method are compared to a perfectly matched layer method. Numerical experiments indicate that the approximation error of the Hardy space decays exponentially in the number of Hardy space modes.
15 citations
01 Jan 2011
10 citations
Cites methods from "Hardy-Raum Methoden zur numerischen..."
...For a more detailed introduction than the one in Chapter 2, we refer to Hohage and Nannen [HN09], [Nan08] and the references therein....
[...]
TL;DR: In this article, a transparent boundary condition based on the pole condition has been proposed for resonance problems, which has one complex tuning parameter and is invariant to variations of this parameter.
Abstract: In many implementations of transparent boundary conditions for resonance problems, spurious modes arise.
We have developed a transparent boundary condition based on the pole condition that has one complex
tuning parameter. Numerical experiments suggest that the artificial eigenvalues are due to badly converged
solutions in the exterior domain and thus are strongly dependent on variations of this parameter while
physical solutions are well converged and thus almost invariant. Hence it is possible to differentiate between
spurious and physical solutions by doing a sensitivity analysis of the eigenvalues.
5 citations
References
More filters
Book•
01 Jan 1966
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Abstract: "The monograph by T Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory
In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4) Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8) The fundamentals of semigroup theory are given in chapter 9 The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10
The first edition is now 30 years old The revised edition is 20 years old Nevertheless it is a standard textbook for the theory of linear operators It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field
Zentralblatt MATH, 836
19,846 citations
TL;DR: Numerical experiments and numerical comparisons show that the PML technique works better than the others in all cases; using it allows to obtain a higher accuracy in some problems and a release of computational requirements in some others.
9,875 citations
Book•
01 Jan 1944
TL;DR: The tabulation of Bessel functions can be found in this paper, where the authors present a comprehensive survey of the Bessel coefficients before and after 1826, as well as their extensions.
Abstract: 1. Bessel functions before 1826 2. The Bessel coefficients 3. Bessel functions 4. Differential equations 5. Miscellaneous properties of Bessel functions 6. Integral representations of Bessel functions 7. Asymptotic expansions of Bessel functions 8. Bessel functions of large order 9. Polynomials associated with Bessel functions 10. Functions associated with Bessel functions 11. Addition theorems 12. Definite integrals 13. Infinitive integrals 14. Multiple integrals 15. The zeros of Bessel functions 16. Neumann series and Lommel's functions of two variables 17. Kapteyn series 18. Series of Fourier-Bessel and Dini 19. Schlomlich series 20. The tabulation of Bessel functions Tables of Bessel functions Bibliography Indices.
9,584 citations
Book•
01 Jan 1978TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Abstract: Preface 1. Elliptic boundary value problems 2. Introduction to the finite element method 3. Conforming finite element methods for second-order problems 4. Other finite element methods for second-order problems 5. Application of the finite element method to some nonlinear problems 6. Finite element methods for the plate problem 7. A mixed finite element method 8. Finite element methods for shells Epilogue Bibliography Glossary of symbols Index.
8,407 citations