We address in this article the computation of the convex solutions of the Dirichlet
problem for the real elliptic Monge − Ampère equation for general convex domains in two
dimensions. The method we discuss combines a least-squares formulation with a relaxation
method. This approach leads to a sequence of Poisson − Dirichlet problems and another
sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite
element approximations with a smoothing procedure are used for the computer implementation
of our least-squares/relaxation methodology. Domains with curved boundaries are easily
accommodated. Numerical experiments show the convergence of the computed solutions to
their continuous counterparts when such solutions exist. On the other hand, when classical
solutions do not exist, our methodology produces solutions in a least-squares sense.