Numerical Methods for the 2-Hessian Elliptic Partial Differential Equation

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.