  In this work we present a numerical method for the Optimal Mass
Transportation problem. Optimal Mass Transportation (OT) is an active research
field in mathematics.It has recently led to significant theoretical results as
well as applications in diverse areas. Numerical solution techniques for the OT
problem remain underdeveloped. The solution is obtained by solving the second
boundary value problem for the MA equation, a fully nonlinear elliptic partial
differential equation (PDE). Instead of standard boundary conditions the
problem has global state constraints. These are reformulated as a tractable
local PDE. We give a proof of convergence of the numerical method, using the
theory of viscosity solutions. Details of the implementation and a fast
solution method are provided in the companion paper arXiv:1208.4870.
