Abstract : The Dirichlet problem for the non-linear elliptic partial differential equation a(x,y,u(x,y))u, sub xx + c(x,y,u(x,y))u, sub yy - gamma(x, y,u(x,y))u = O is studied. It is assumed that the coefficients are strictly positive and Lipschitz in the argument u(x,y). It is then proved that the solution may be uniformly approximated by the solution to the associated difference equation provide ed that a certain inequality, relating bounds on the coefficients, is satisfied.