Title: Laplace in the vicinity of corners


In classical potential theory, elliptic partial differential equations are reduced to second kind boundary integral equations by representing the solutions to the differential equations by single-layer or double-layer potentials on the boundaries of the regions. After discretization, the resulting linear systems are generally better-conditioned than direct discretization of the differential equation. For regions with smooth boundaries there exist a variety of methods, both direct and iterative, for solving these linear systems quickly and with high precision. However, near corners the solutions to both the differential and integral equations have singularities which pose significant challenges to many existing approaches. In this talk I will describe a class of algorithms for the solution of Laplace's equation on polygonal domains with Dirichlet and Neumann boundary conditions. In particular, I will describe a high-order solver for Laplace's equation on polygonal domains requiring relatively few degrees of freedom to resolve the behaviour near corners accurately. Speaker(s): Jeremy Hoskins (Yale University)