We describe the construction of a collection of quadrature formulae suitable for the efficient discretization of certain boundary integral equations on a very general class of two-dimensional domains with corner points. The resulting quadrature rules allow for the rapid high accuracy solution of Laplace's equation and the Helmholtz equation on such domains. Our approach can be adapted to many other boundary value problems as well as to the case of surfaces with singularities in three dimensions. The performance of the quadrature rules is illustrated with several numerical examples.

No abstract