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The Steklov spectrum of the Helmholtz operator

Resource type
Thesis type
(Thesis) M.Sc.
Date created
2023-12-13
Authors/Contributors
Abstract
The Steklov eigenvalue problem is a boundary value problem where the spectral parameter relates the Dirichlet and Neumann traces. It has been extensively studied for the Laplacian. In this thesis, we study the Steklov problem for the Helmholtz operator −∆ − µ2 with real wave number µ for bounded planar domains. After reformulating this problem as a boundary integral equation we implement a numerical method with the help of layer potentials approximated using appropriate quadratures. We obtain a generalized eigenvalue problem for the Steklov-Helmholtz eigenvalues and eigendensities, where the eigenfunctions are reconstructed using the single layer potential. We describe a numerical approach for the reformulated problem. We observe exponential convergence of the numerically computed Steklov-Helmholtz eigenvalues. A Weyl-type law is observed for the asymptotics of the spectrum. We report on additional experiments including a shape optimization problem. The numerical scheme is tested on a variety of domains of genus 0 and 1. Our approach can be adapted to other problems as well.
Document
Extent
115 pages.
Identifier
etd22850
Copyright statement
Copyright is held by the author(s).
Permissions
This thesis may be printed or downloaded for non-commercial research and scholarly purposes.
Supervisor or Senior Supervisor
Thesis advisor: Nigam, Nilima
Thesis advisor: Sun, Weiran
Language
English
Member of collection
Download file Size
etd22850.pdf 20.95 MB

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