# Numerical methods for fully nonlinear elliptic equations

### Roland Glowinski

University of Houston, USA

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## Abstract

In this article we address the numerical solution of fully nonlinear elliptic equations, a typical one being the canonical Monge–Ampère equation det **D**2ψ = f, where **D**2ψ is the Hessian of the unknown function **D**2ψ, 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 **D**2ψ, and *α* > 1. The results of numerical experiments will be presented.