Numerical methods for fully nonlinear elliptic equations

Abstract

In this article we address the numerical solution of fully nonlinear elliptic equations, a typical one being the canonical Monge–Ampère equation det D2ψ = f, where D2ψ is the Hessian of the unknown function D2ψ, and where f is a given positive function. We will start ourdiscussionby a short review of methods which have been used by various authors for the solution of fully nonlinear elliptic equations. Then we will give a relatively detailed description of one of these approaches when applied to the solution of the above Monge–Ampère equation and of the following two-dimensional Pucci’s equation α λ+ + λ- = 0, where λ+ and λ- are, respectively, the largest and smallest eigenvalues of D2ψ, and α > 1. The results of numerical experiments will be presented.