The discretization process of the full potential equation (FPE) both in the quasi-linear and in the conservation form, is addressed. This work introduce the rst stage toward a development of a fast and ecient FPE solver, which is based on the algebraic multigrid (AMG) method. 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). A pointwise relaxation method when applied directly to the upwind discrete operator, in the supersonic ow regime, is unstable. Resolving this diculty 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 rst order downwind operator. This new operator is designed in such a way that makes the pointwise relaxation applied to the product operator to become stable. The discretization of the FPE in the conservation form is based on the body-tted structured grid approach. In addition the 2D stable operator in the supersonic ow regime was extended to 3D case. We present a 3D pointwise relaxation procedure that is stable both in the subsonic and supersonic ow regimes. This was veried by the Von-Neumann stability analysis.