The objective of this work is to develop a highly efficient solver for the Full Potential Equation (FPE) that will be able to compute transonic external and internal flows attaining a (nearly) linear computational complexity. The key innovation of this work is in the solver's efficiency and in the fact that it is achieved by means of adapting and applying the algebraic multi-grid (AMG) approach to solving the problem. The mathematical difficulties of the problem are associated with the fact that the governing equation changes its type from elliptic (subsonic ow) to hyperbolic (supersonic ow). A pointwise relaxation method when applied directly to the upwind discrete operator, in the supersonic ow regime, is unstable. Resolving this difficulty is the main achievement of this work. A stable pointwise direction independent relaxation was developed for the supersonic and subsonic ow regimes. This stable relaxation is obtained by post-multiplying the original operator by a certain simple first order downwind operator. This new operator is designed in such a way that the pointwise relaxation applied to the product operator becomes stable. A variety of issues regarding the AMG coarsening and construction of transfer operators is addressed in order to achieve the required efficiency for the problems under consideration. An improved coarsening process was developed. Instead of using a fixed threshold parameter in order to select the coarse-grid points, we developed a dynamic threshold parameter as a measure of the strength of connection between the matrix variables. The coarsening by dynamic threshold was shown to be less effective for certain elliptic problems (subsonic flow), but for supersonic flow regime where the operator does not form an M-matrix, we obtained much better performance. In some cases where an irregular grid, shock waves, and extreme nonlinearity are involved, the dynamic threshold is more than necessary in order to achieve convergence. A modified formulation of the interpolation operator is presented. While the standard interpolation is suitable mainly for problems that are characterized by M-matrix form,the proposed formula is more accurate and can be used for more general matrix problems. The proposed interpolation operator includes the choice of negative weights, which is necessary in some cases. In addition, the FMG approach in the context of AMG was developed as a tool to deal with a nonlinear problems... (truncated by viXra Admin to < 400 words).