Two-Grid Discretization for Finite Element Approximations of the Elliptic Monge-Ampere Equation

We consider the C0 interior penalty and mixed finite element approximations of the Monge-Ampère equation with C0 Lagrange elements. We solve the discrete nonlinear system of equations with a two-grid method. This method consists in first solving the nonlinear problem on a coarse grid, and then using that solution as the initial guess for a single Newton iteration on a fine grid. Numerical results demonstrate that the two-grid method is more efficient than Newton's method on the fine grid. We give new proofs of convergence for each discrete problem, and prove the convergence of the two-grid methods with optimal error estimates in each case. We give the first theoretical study of multi-grid methods for finite element discretizations of the Monge-Ampère equation. Finally, we prove convergence of a time marching method for solving the nonlinear system resulting from the C0 interior penalty discretization.