Solving Laplace's equation in a polygon

There is no more classical problem of numerical PDE than the Laplace equation in a polygon, but Abi Gopal and I think we are on to a big step forward. The traditional approaches would be finite elements, giving a 2D representation of the solution, or integral equations, giving a 1D representation. The new approach, inspired by an approximation theory result of Donald Newman in 1964, leads to a "0D representation" -- the solution is the real part of a rational function with poles clustered exponentially near the corners of the polygon. The speed and accuracy of this approach are remarkable. For typical polygons of up to 8 vertices, we can solve the problem in less than a second on a laptop and evaluate the result in a few microseconds per point, with 6-digit accuracy all the way up to the corner singularities. We don't think existing methods come close to such performance. Next step: Helmholtz?