We address in this article the computation of the convex solutions of the Dirichletproblem for the real elliptic Monge − Ampère equation for general convex domains in twodimensions. The method we discuss combines a least-squares formulation with a relaxationmethod. This approach leads to a sequence of Poisson − Dirichlet problems and anothersequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finiteelement approximations with a smoothing procedure are used for the computer implementationof our least-squares/relaxation methodology. Domains with curved boundaries are easilyaccommodated. Numerical experiments show the convergence of the computed solutions totheir continuous counterparts when such solutions exist. On the other hand, when classicalsolutions do not exist, our methodology produces solutions in a least-squares sense.

The sin2ψ method can be formulated as a single system of simultaneous linear equations. Using this it is easy to show that the sin2ψ method is not a least-squares method. It further helps to compare the accuracies of the stress tensors obtained by the sin2ψ method and the method of least squares. Quantitative comparisons have been made for different fictitious measurements. It is shown that the unnecessary loss in accuracy by using the sin2ψ method is quite significant and by no means negligible. The same course of action has been applied to compare the so-called Dölle-Hauk method with a least-squares method; the result is similar. Some other methods for X-ray stress determination, most often similar to the sin2ψ method, and their shortcomings are also discussed briefly, together with the corresponding, more effective and more versatile least-squares method.