CSC Weekly Seminar
TASC-2, Rm 8500
2:30 pm

Speaker: EldarAkhmetgaliyev, Caltech


Title: Integral equation methods for singular problems and application to
the evaluation of Laplace eigenvalues.

Abstract:
Boundary integral equation methods represent a powerful tool for
the numerical solution of a variety of problems in acoustics,
electromagnetics, fluid mechanics etc.
Together with Oscar Bruno at Caltech, we developed a numerical method
for the solution of scattering problems with mixed Dirichlet/Neumann
boundary conditions. We investigated and obtained the exact form of the
solution singularities -- which arise
at transition points where Dirichlet and Neumann
boundary conditions meet. These singularities are incorporated in the
numerical approach and are resolved via Fourier Continuation technique.
The resulting method exhibits spectral convergence.
Additionally, jointly with Nilima Nigam at SFU, we applied the mixed
boundary value
solver to Laplace eigenvalue problems. The challenge is in eigenvalue search
as a result of the properties of the objective function, which requires
scanning through a range of frequencies. We introduce an improved search
algorithm, that allows to locate the eigenvalues using standard
root-finding methods.
We also apply another integral equation method for domains with corners
for a mode matching problem in electromagnetics. Jointly with Ahmed
Akgiray at Caltech, we calculated TE and TM modes for domains of
specific geometry. Those are further used for antenna design.