A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation

In this paper, we study finite element approximations of the viscosity solution of the fully nonlinear Monge-Ampère equation, det(Du) = f (> 0) using the well-known nonconforming Morley element. Our approach is based on the vanishing moment method, which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and Feng in [15]. The vanishing moment method approximates the Monge-Ampère equation by the fourth order quasilinear equation − ∆u + det(Du ) = f with appropriate boundary conditions. We develop a finite element scheme using the n−dimensional Morley element introduced in [27] to approximate the regularized fourth order problem in two and three dimensions, and then derive optimal order error estimates.