LAPLACE EQUATION IN A POLYGON

Consider the Laplace equation in a polygon with continuous Dirichlet boundary data.  One could compute the solution u with finite elements, based on a two-dimensional representation of the solution, or integral equations, based on a one-dimensional representation.  We propose a “zero-dimensional”

representation: u is the real part of a rational function with poles exponentially clustering near each vertex.

Thanks to an effect first identified by Donald Newman in 1964, the convergence is root-exponential as a function of the number of degrees of freedom, i.e. of the form exp(-C*sqrt(N)) with C>1.  In practice, with 40 lines of Matlab code, we can solve problems with 3-8 vertices in a second of laptop time, with 8-digit accuracy all the way up to the singularities in the corners.  Evaluation of the solution takes around 20 microseconds per point.  This is joint work with Abi Gopal.