Asymptotic analysis of boundary integral equations of regions with high curvature

In this project we investigate the behavior of layer potentials in regions of high curvature in two dimensions, in particular an asymptotically collapsing ellipse. Layer potentials arise in boundary integral methods and offer several advantages numerically but can be affected by regions of high curvature. Such phenomena appear in slender body theory. In this thesis we propose two approaches to address this challenge. We propose a modification of quadrature rules using asymptotic methods, and a spectral method when one can find the analytic Fourier coefficients. We apply these techniques to several problems: Laplace's problems (interior Dirichlet and exterior Neumann), and a scattering problem