Spring 2016

Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall II, unless noted otherwise. Refreshments are served at 11:30 am. For questions about the seminar schedule, please contact Yassine Boubendir.


Date: April 15, 2016

Speaker: Nilima Nigam
Department of Mathematics,
Simon Fraser University

University Profile

Title: "High Accuracy Computation of Mixed Dirichlet-Neumann Eigenvalues"

Abstract:

Eigenfunctions of the Laplace operator with mixed Dirichet-Neumann boundary conditions may possess singularities, especially if the Dirichlet-Neumann junction occurs at angles $\geq \frac{\pi}{2}$. This suggests the use of boundary integral strategies to solve such eigenproblems. As with boundary value problems, integral-equation methods allow for a reduction of dimension, and the resolution of singular behaviour which may otherwise present challenges to volumetric methods.

In this talk, we present a novel integral-equation algorithm for mixed Dirichlet-Neumann eigenproblems. This is based on joint work with Oscar Bruno and Eldar Akhmetgaliyev (Caltech).

For domains with smooth boundary, the singular behaviour of the eigenfunctions at Dirichlet-Neumann junctions is incorporated as part of the discretization strategy for the integral operator. The discretization we use is based on the high-order Fourier Continuation method (FC).

For non-smooth (Lipschitz) domains an alternative high-order discretization is presented which achieves high-order accuracy on the basis of graded meshes.

In either case (smooth or Lipschitz boundary), eigenvalues are evaluated by examining the minimal singular values of a suitably stabilized discretesystem. This is in the spirit of the modification proposed by Trefethen and Betcke in the modified method of particular solutions.

The method is conceptually simple, and allows for highly accurate and efficient computation of eigenvalues and eigenfunctions, even in challenging geometries.