Iterative methods for $k$-Hessian equations

On a domain of the n-dimensional Euclidean space, and for an integer $k = 1, \dotsc , n$, the $k$-Hessian equations are fully nonlinear elliptic equations for $k \gt 1$ and consist of the Poisson equation for $k = 1$ and the Monge–Ampère 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–Ampère equation is established and new Gauss–Seidel type iterative methods for $2$-Hessian equations are introduced.

Keywords

$k$-Hessian, discrete Schauder estimates, finite difference

2010 Mathematics Subject Classification

Primary 65N06. Secondary 35J60, 65N12.

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The author was supported in part by NSF grants DMS-0811052 and DMS-1319640, and by the Sloan Foundation.

Received 4 April 2017

Accepted 14 June 2018

Published 24 July 2018