ON THE LEAST-SQUARES SOLUTION OF THE DIRICHLET PROBLEM FOR THE ELLIPTIC MONGE-AMP ` ERE EQUATION IN DIMENSION TWO

We address the numerical solution of the Dirichlet problem for the real elliptic Monge- Ampere equation for arbitrary domains in two dimensions. The numerical 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 exact counterparts when such solutions exist. On the other hand, when smooth solutions do not exist, our least-squares based methodology produces generalized solutions which can be viewed as viscosity solutions, but in a sense different from Ishii & Lions'. Resume. Nousdans cet article la resolution numerique de l'´equation de Monge-Ampere elliptique dans des domaines de forme arbitraire en deux dimensions. Une methode de moindres carres est coupleeun algorithme de relaxation, conduisantla resolution d'une suite de problemes variation- nels lineaires, et d'une suite de problemes de valeurs propres en deux dimensions. Une approximation parelements finis mixtes coupleeune methode de regularisation est utilisee, de sorte que les domaines avec frontiere courbe sont traites facilement. Des experiences numeriques montrent l'efficacite de la methode, ainsi que des bonnes proprietes de convergence.