Solution Methods for $C^0$ Interior Penalty and Mixed Finite Element Discretizations of the Elliptic Monge-Ampère Equation
Abstract: The Monge-Ampère equation plays a role in applications in geometry, optics, and other areas. The most natural finite element discretization of the Dirichlet problem for the Monge-Ampère equation is with $C^1$ elements, which lead to unruly discrete systems. We discretize the problem with a $C^0$ interior penalty method, allowing the use of simpler $C^0$ Lagrange elements. We then prove convergence of two methods for resolving the nonlinear discrete problem: an iterative time marching method that may capture more accurate results for nonsmooth solutions than does Newton's method, and a two-grid method which is computationally more efficient than Newton's method.