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.