Coulomb interactions on planar structures: Inverting the square root of the Laplacian

We present an adaptive fast multipole method for inverting the square root of the Laplacian in two dimensions. Solving this problem is the dominant co mputational cost in many applications arising in electrical engineering, ge ophysical fluid dynamics, and the study of thin lms. It corresponds to the evaluation of the field induced by a planar distribution of charge or vorti city. Our algorithm is direct and assumes only that the source distribution is discretized using an adaptive quad-tree. The amount of work grows linea rly with the number of mesh points.