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Fast Numerical Methods for Mixed, Singular Helmholtz Boundary Value Problems and Laplace Eigenvalue Problems - with Applications to Antenna Design, Sloshing, Electromagnetic Scattering and Spectral Geometry

Citation

Akhmetgaliyev, Eldar (2016) Fast Numerical Methods for Mixed, Singular Helmholtz Boundary Value Problems and Laplace Eigenvalue Problems - with Applications to Antenna Design, Sloshing, Electromagnetic Scattering and Spectral Geometry. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z97P8W93. http://resolver.caltech.edu/CaltechTHESIS:08202015-162838809

Abstract

This thesis presents a novel class of algorithms for the solution of scattering and eigenvalue problems on general two-dimensional domains under a variety of boundary conditions, including non-smooth domains and certain "Zaremba" boundary conditions - for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the methods for the Zaremba problems on smooth domains concern detailed information, which is put forth for the first time in this thesis, about the singularity structure of solutions of the Laplace operator under boundary conditions of Zaremba type. The new methods, which are based on use of Green functions and integral equations, incorporate a number of algorithmic innovations, including a fast and robust eigenvalue-search algorithm, use of the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities, and newly derived quadrature rules which give rise to high-order convergence even around singular points for the Zaremba problem. The resulting algorithms enjoy high-order convergence, and they can tackle a variety of elliptic problems under general boundary conditions, including, for example, eigenvalue problems, scattering problems, and, in particular, eigenfunction expansion for time-domain problems in non-separable physical domains with mixed boundary conditions.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:applied mathematics, integral equations, Laplace eigenvalues
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Awards:The W.P. Carey & Co., Inc., Prize in Applied and Computational Mathematics, 2016
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Bruno, Oscar P.
Thesis Committee:
  • Bruno, Oscar P. (chair)
  • Beck, James L.
  • Simon, Barry M.
  • Pullin, Dale Ian
Defense Date:8 July 2015
Record Number:CaltechTHESIS:08202015-162838809
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:08202015-162838809
DOI:10.7907/Z97P8W93
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9110
Collection:CaltechTHESIS
Deposited By: Eldar Akhmetgaliyev
Deposited On:02 Sep 2015 16:45
Last Modified:18 May 2017 17:21

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