An implementation of the Geometric Multi-Grid (GMG) method, with full approximation scheme/storage (FAS) algorithm, in a numerical study of steady Buoyancy-driven convection with axis-symmetric flows in vertical cylinders is presented. The particular system examined is a cylinder with an aspect ratio a (radius divided by height) of 4. The fluid is heated from below and cooled from above, and the circular wall of the vessel is insulated. The Rayleigh (Ra) and fluid Prandtl (Pr) numbers are respectively 5 6.4 10 and 7. A non-linear system of equations was formulated in stream function-vorticity-temperature variables and discretized using a monotonic conservative finite difference scheme of second order accuracy. The steady state condition was solved for purposes of comparing two numerical methods: the GMG FAS and the GaussSeidel method with lexicographic ordering (GS-LEX). The GMG FAS method has significantly higher efficiency in CPU performance compared to the pure GS-LEX method for fine grids only if the tolerance value for stopping iteration process is chosen not too small. A procedure for the selection of adjustment parameters for GMG FAS algorithm is proposed and tested for different grid sizes. Details regarding convergence criteria are addressed.