Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions
This provides theoretical justification for using standard finite elements on a challenging PDE, but the result is incremental as it builds on existing discretization techniques.
The paper proves convergence of a standard finite element discretization for the elliptic Monge-Ampère equation, extending previous results from smooth to non-smooth Aleksandrov solutions, thereby providing theoretical validation for non-monotone finite element methods.
We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge-Ampere equation using finite dimensional spaces of piecewise polynomial C0 or C1 functions. Standard discretizations of the type considered in this paper have been previous 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.