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

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.

Not just another pretty reef: the Gainesville Florida Reef, a satellite of the worldwide Hyperbolic Crochet Coral Reef project: poster presentation

The Gainesville Florida Reef, a satellite of the Worldwide Hyperbolic Crochet Coral Reef, project not only shows the beauty of reefs but serves to: • Foster scientific communication through the visual arts • Raise awareness of the fragility of our coral reefs and the entire ecosystem • Support learning by creating physical models of geometric principles • Connect several areas on campus, including fine arts, mathematics and ecology and environmental sciences through collaboration and mutual interest • Encourage local community and alumni involvement through creating, observing and learning