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On spurious solutions in finite element approximations of resonances in open systems
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. (UMIT)
2017 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 74, no 10, p. 2385-2402Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Elsevier, 2017. Vol. 74, no 10, p. 2385-2402
Keywords [en]
Scattering resonances, Lippmann–Schwinger equation, Nonlinear eigenvalue problems, Acoustic resonator, Dielectric resonator, Bragg resonator
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-138096DOI: 10.1016/j.camwa.2017.07.020ISI: 000415908400013OAI: oai:DiVA.org:umu-138096DiVA, id: diva2:1130348
Funder
Swedish Research Council, 621-2012-3863Available from: 2017-08-09 Created: 2017-08-09 Last updated: 2019-05-20Bibliographically approved
In thesis
1. Reliable hp finite element computations of scattering resonances in nano optics
Open this publication in new window or tab >>Reliable hp finite element computations of scattering resonances in nano optics
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Eigenfrequencies are commonly studied in wave propagation problems, as they are important in the analysis of closed cavities such as a microwave oven. For open systems, energy leaks into infinity and therefore scattering resonances are used instead of eigenfrequencies. An interesting application where resonances take an important place is in whispering gallery mode resonators.

The objective of the thesis is the reliable and accurate approximation of scattering resonances using high order finite element methods. The discussion focuses on the electromagnetic scattering resonances in metal-dielectric nano-structures using a Drude-Lorentz model for the description of the material properties. A scattering resonance pair satisfies a reduced wave equationand an outgoing wave condition. In this thesis, the outgoing wave condition is replaced by a Dirichlet-to-Neumann map, or a Perfectly Matched Layer. For electromagnetic waves and for acoustic waves, the reduced wave equation is discretized with finite elements. As a result, the scattering resonance problem is transformed into a nonlinear eigenvalue problem.

In addition to the correct approximation of the true resonances, a large number of numerical solutions that are unrelated to the physical problem are also computed in the solution process. A new method based on a volume integral equation is developed to remove these false solutions.

The main results of the thesis are a novel method for removing false solutions of the physical problem, efficient solutions of non-linear eigenvalue problems, and a new a-priori based refinement strategy for high order finite element methods. The overall material in the thesis translates into a reliable and accurate method to compute scattering resonances in physics and engineering.

Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2019. p. 35
Series
Research report in mathematics, ISSN 1653-0810 ; 67
Keywords
Scattering resonances, Helmholtz problems, pseudospectrum, Lippmann-Schwinger equation, finite element methods, nonlinear eigenvalue problems, spurious solutions
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-159154 (URN)978-91-7855-076-0 (ISBN)
Public defence
2019-06-13, MA121, MIT-huset, Umeå, 13:00 (English)
Opponent
Supervisors
Available from: 2019-05-23 Created: 2019-05-20 Last updated: 2019-05-21Bibliographically approved

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Araujo-Cabarcas, Juan CarlosEngström, Christian

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