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(D2u) = 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 (J Sci Comput 38(1):74–98, 2009). The vanishing moment method approximates the Monge-Ampère equation by the fourth order quasilinear equation $${-\epsilon\Delta^2u^\epsilon + {\rm det}(D^2u^\epsilon) = f}$$ with appropriate boundary conditions. We develop a finite element scheme using the n-dimensional Morley element introduced in Wang and Xu (Numer Math 103:155–169, 2006) to approximate the regularized fourth order problem in two and three dimensions, and then derive optimal order error estimates.

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