CRM-McGill Applied Math Seminar
Tuesday April 8, 2008, 3:35pm
McGill, Burnside Hall Room 1205

Michael Haslam
York University

Title: Fast High-Order Integral Equation Methods (with application
to optical gratings and antenna design) 

Abstract:
Partial differential equations are often reformulated as integral
equations, via, say, Green's theorem. Such a reformulation can be highly
advantageous from a numerical point of view, since integral formulations
make use of domain-termination techniques unnecessary, they can be used in
conjunction with high-order numerical integration schemes, and they
completely bypass difficulties associated with numerical differentiation.
In this talk, we discuss the high-order solution of both first- and
second-kind integral equations whose kernels contain logarithmic
singularities.  Our schemes rely on the exact treatment of the singular
term by expressing the kernel in the form G(z) = F1(z)ln|z| + F_2(z),
where F1(z) and F_2(z) are complex functions of the real variable z.  For
F1 and F2 analytic, the algorithms we propose converge
super-algebraically: faster than O(1/N^m) and O(1/M^m) for any positive
integer m, where N and M are the numbers of unknowns and the number of
integration points required for construction of the discretized operator,
respectively. We illustrate the effectiveness of our methods with
applications to wire antenna problems and deep optical gratings containing
geometric singularities. (With Oscar Bruno, Caltech)