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. https://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): |
|
Thesis Committee: |
|
Defense Date: | 8 July 2015 |
Record Number: | CaltechTHESIS:08202015-162838809 |
Persistent URL: | https://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: | 04 Oct 2019 00:09 |
Thesis Files
|
PDF
- Updated Version
See Usage Policy. 28MB |
Repository Staff Only: item control page