Standard Finite Elements for the Numerical Resolution of the Elliptic Monge-ampère Equation: Aleksandrov Solutions

We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge-Ampère equation using finite dimensional spaces of piecewise polynomial C functions. Discretizations of the type considered in this paper have been previously analyzed in the case the equation has a smooth solution and numerous numerical evidence of convergence were given in the case of non smooth solutions. Our convergence result is valid for non smooth solutions, is given in the setting of Aleksandrov solutions, and consists in discretizing the equation in a subdomain with the boundary data used as an approximation of the solution in the remaining part of the domain. Our result gives a theoretical validation for the use of a non monotone finite element method for the Monge-Ampère equation.