Numerical methods for the 2-Hessian elliptic partial differential equation

The elliptic 2-Hessian equation is a fully nonlinear partial differential equation (PDE) that is related to intrinsic curvature for three-dimensional manifolds. We ntroduce 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 iscretizations are obtained using Newton's method. Computational results are presented on a number of exact solutions which range in regularity from mooth to nondifferentiable and in shape from convex to nonconvex.