ABSTRACT: In this talk I will present recent results on analyticity of
solutions to free boundary and boundary value problems with respect to
variations of their domains of definition, and the application of
these results to numerical approximation techniques. Specifically,
I will show how the analysis can be used to design high-order
boundary perturbation schemes with substantially improved conditioning
properties over classical perturbation methods. As I will demonstrate,
this enhanced stability can be combined with methods of analytic
continuation to overcome some of the limitations of traditional
approaches. I shall illustrate these results through applications to
some classical (boundary value) problems of electromagnetic scattering
and (free-boundary) models for the evolution of surface gravity water
waves.