This article reports the development of an ecient, and robust full potential equation (FPE) solver for transonic ow problems, which is based on the algebraic multigrid (AMG) method. AMG method solves algebraic systems based on multigrid principles but in a way that it is independent on the problem's geometry. The mathematical diculties of the problem are associated with the fact that the governing equation changes its type from elliptic (subsonic ow) to hyperbolic (supersonic ow). The ow solver is based of the body-tted structured grid approach in complex geometries. We demonstrate the AMG performance on various model problems with dierent ow speed from subsonic to transonic conditions. The computational method was demonstrated to be capable of predicting the shock formation and achieving residual reduction of roughly an order of magnitude per cycle, both for elliptic and hyperbolic problems.