Results obtained by using a fast Poisson solver to accelerate the rate of convergence of the iterative scheme are described. A fast iterative method for transonic flow calculations is formulated first, and then application is made to the transonic potential flow equation in a mapped domain. It is concluded that the use of a fast elliptic solver in combination with relaxation is an effective way to accelerate the convergence of transonic flow calculations, particularly when a marching scheme can be used to treat the supersonic zone in the relaxation process. 3 figures (RWR)