arXiv subjects

Christian Engström

Publications and source records attributed to Christian Engström.

8 recordsLinked to original sources

arXiv2020

Spectra of Gurtin-Pipkin type of integro-differential equations and applications to waves in graded viscoelastic structures

In this paper, we study spectral properties and spectral enclosures for the Gurtin-Pipkin type of integro-differential equations in several dimensions. The analysis is based on an operator function and we consider the relation between the studied operator function and other formulations of the spectral problem. The theory is applied to wave equations with Boltzmann damping.

math-ph↗

arXiv2020

On spurious solutions encountered in Helmholtz scattering resonance computations in R^d with applications to nano-photonics and acoustics

In this paper, we consider a sorting scheme for potentially spurious scattering resonant pairs in one- and two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a Lippmann-Schwinger type of volume integral equation and can, therefore, be applied to structures with graded materials as well as to configurations including piece-wise constant material properties. For TM/TE polarized electromagnetic waves and for acoustic waves, we compute first approximations of scattering resonances with finite elements. Then, we apply the novel sorting scheme to the computed eigenpairs and use it to mark potentially spurious solutions in electromagnetic and acoustic scattering resonances computations at a low computational cost. Several test cases with Drude-Lorentz dielectric resonators as well as with graded material properties are considered.

math-ph↗

arXiv2019

Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures

In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel $hp$-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS. Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform $h$ and $p$ strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The $hp$-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.

math.NA↗

arXiv2018

Accumulation of Complex Eigenvalues of a Class of Analytic Operator Functions

For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These results are used to prove sufficient conditions for eigenvalue accumulation to the poles and to infinity of rational operator functions. Finally, an application of electromagnetic field theory is given.

math.FA↗

arXiv2017

On Equivalence and Linearization of Operator Matrix Functions with Unbounded Entries

In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a method of finding equivalences and linearizations to a general case of operator matrix polynomials is presented.

math.FA↗

arXiv2017

Enclosure of the Numerical Range of a Class of Non-Selfadjoint Rational Operator Functions

In this paper we introduce an enclosure of the numerical range of a class of rational operator functions. In contrast to the numerical range the presented enclosure can be computed exactly in the infinite dimensional case as well as in the finite dimensional case. Moreover, the new enclosure is minimal given only the numerical ranges of the operator coefficients and many characteristics of the numerical range can be obtained by investigating the enclosure. We introduce a pseudonumerical range and study an enclosure of this set. This enclosure provides a computable upper bound of the norm of the resolvent.

math.FA↗

arXiv2017

On spurious solutions in finite element approximations of resonances in open systems

In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann-Schwinger equation and we use computations of the pseudospectrum to show that this is a suitable choice. Numerical simulations indicate that the presented test can distinguish between spurious eigenvalues and true eigenvalues also in difficult cases. Keywords: scattering resonances, Lippmann-Schwinger equation, nonlinear eigenvalue problems, acoustic resonator, dielectric resonator, Bragg resonator

math.NA↗

arXiv2015

Non-linear eigenvalue problems and applications to photonic crystals

We establish new analytic and numerical results on a general class of rational operator Nevanlinna functions that arise e.g. in modelling photonic crystals. The capability of these dielectric nano-structured materials to control the flow of light depends on specific features of their eigenvalues. Our results provide a complete spectral analysis including variational principles and two-sided estimates for all eigenvalues along with numerical implementations. They even apply to multi-pole Lorentz models of permittivity functions and to the eigenvalues between the poles where classical min-max variational principles fail completely. In particular, we show that our abstract two-sided eigenvalue estimates are optimal and we derive explicit bounds on the band gap above a Lorentz pole. A high order finite element method is used to compute the two-sided estimates of a selection of eigenvalues for several concrete Lorentz models, e.g. polaritonic materials and multi-pole models.

math-ph↗