A Finite Element/Operator-Splitting Method for the Numerical Solution of the Three Dimensional Monge–Ampère Equation

In the present article we extend to the three-dimensional elliptic Monge–Ampère equation the method discussed in Glowinski et al. (J Sci Comput 79:1–47, 2019) for the numerical solution of its two-dimensional variant. As in Glowinski et al. (2019) we take advantage of an equivalent divergence formulation of the Monge–Ampère equation, involving the cofactor matrix of the Hessian of the solution. We associate with the above divergence formulation an initial value problem, well suited to time discretization by operator splitting and space approximation by low order mixed finite element methods. An important ingredient of our methodology is forcing the positive semi-definiteness of the approximate Hessian by a hard thresholding eigenvalue projection. The resulting method is robust and easy to implement. It can handle problems with smooth and non-smooth solutions on domains with curved boundary. Using piecewise affine approximations for the solution and its six second-order derivatives, one can achieve second-order convergence rates for problems with smooth solutions.