Title: Quadrature for layer potentials with near singularities

Boundary integral methods provide an efficient way of solving elliptic partial differential equations, by representing the solution as a layer potential. However, efficient evaluation of the solution close to the boundary of the domain presents a significant problem. The root cause of this is the fact that common quadrature methods for smooth integrands, such as Gauss-Legendre quadrature and the trapezoidal rule, are ill-suited for integrands that have a singularity near the interval of integration. The nature of this error is however well understood, and I will show how it can be approximated numerically. I will also review two relatively recent methods for evaluating layer potentials near a boundary; one uses the smoothness of the solution to form a local expansion of the solution (QBX), the other computes modified quadrature weights using a complex variable formulation. These methods can be in turn be improved, a least in some cases, by applying the techniques developed for the numerical approximation of quadrature errors.