Iterative methods for k-Hessian equations
This work provides theoretical and numerical advances for solving k-Hessian equations, which are important in fully nonlinear elliptic PDEs, but the results are incremental as they extend existing methods to a broader class.
The paper analyzes a 9-point finite difference scheme for k-Hessian equations, proving local uniqueness and quadratic convergence for smooth solutions, and introduces new iterative methods that work for non-smooth solutions, including Gauss-Seidel type methods for 2-Hessian equations.
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.