Standard implementations of the fast multipole method, which compute fields due to point sources or dipoles, cannot be used to accurately evaluate the single- and double-layer potentials of potential theory close to the boundary, or on the boundary when the boundary curves back on itself. We describe the modifications necessary to accurately evaluate layer potentials in two dimensions, which include quadrature rules for the short-range contributions to the field, continuous multipole moments for long-range contributions, and a more complex bookkeeping procedure. We give formulae for second-, third-, and fourth-order methods. We show tests to verify the correctness of the method and numerical results which demonstrate the usefulness of the method for evaluating layer potentials near the boundary.
Mckenney, A. (1996). An adaptation of the fast multipole method for evaluating layer potentials in two dimensions. Computers and Mathematics with Applications, 31(1), 33–57. https://doi.org/10.1016/0898-1221(95)00181-W