Numerical Methods for the 2-Hessian Elliptic Partial Differential Equation

TL;DR

This paper introduces two numerical methods for solving the 2-Hessian elliptic PDE, one with proven convergence and another with higher accuracy but unproven convergence, demonstrated through computational experiments.

Contribution

The paper presents two novel numerical methods for the 2-Hessian elliptic PDE, including a provably convergent scheme and a more accurate, practically convergent scheme without proof.

Findings

The first method is provably convergent to the viscosity solution.

The second method achieves higher accuracy in practice.

Numerical results cover solutions from smooth to nondifferentiable and convex to non-convex shapes.

Abstract

The elliptic 2-Hessian equation is a fully nonlinear partial differential equation (PDE) that is related to intrinsic curvature for three dimensional manifolds. We introduce two numerical methods for this PDE: the first is provably convergent to the viscosity solution, and the second is more accurate, and convergent in practice but lacks a proof. The PDE is elliptic on a restricted set of functions: a convexity type constraint is needed for the ellipticity of the PDE operator. Solutions with both discretizations are obtained using Newton's method. Computational results are presented on a number of exact solutions which range in regularity from smooth to nondifferentiable and in shape from convex to non convex.