Iterative methods for k-Hessian equations

TL;DR

This paper develops and analyzes iterative methods for solving k-Hessian equations, including a finite difference scheme with quadratic convergence and new Gauss-Seidel type methods, effective for both smooth and non-smooth solutions.

Contribution

It introduces a finite difference scheme with proven quadratic convergence and novel Gauss-Seidel type iterative methods for k-Hessian equations, expanding computational tools for these nonlinear PDEs.

Findings

Finite difference scheme has quadratic convergence.

Iterative methods work for non-smooth solutions.

Connection established with Gauss-Seidel method for Monge-Ampere.

Abstract

On a domain of the n-dimensional Euclidean space, and for an integer k=1,...,n, the k-Hessian equations are fully nonlinear elliptic equations for k >1 and consist of the Poisson equation for k=1 and the Monge-Ampere equation for k=n. We analyze for smooth non degenerate solutions a 9-point finite difference scheme. We prove that the discrete scheme has a locally unique solution with a quadratic convergence rate. In addition we propose new iterative methods which are numerically shown to work for non smooth solutions. A connection of the latter with a popular Gauss-Seidel method for the Monge-Ampere equation is established and new Gauss-Seidel type iterative methods for 2-Hessian equations are introduced.