  A new method is introduced for solving Laplace problems on 2D regions with
corners by approximation of boundary data by the real part of a rational
function with fixed poles exponentially clustered near each corner. Greatly
extending a result of D. J. Newman in 1964 in approximation theory, we first
prove that such approximations can achieve root-exponential convergence for a
wide range of problems, all the way up to the corner singularities. We then
develop a numerical method to compute approximations via linear least-squares
fitting on the boundary. Typical problems are solved in < 1s on a laptop to
8-digit accuracy, with the accuracy guaranteed in the interior by the maximum
principle. The computed solution is represented globally by a single formula,
which can be evaluated in tens of microseconds at each point.
