Abstract
The elliptic 2-Hessian equation is a fully nonlinear partial differential equation that appears in geometric surface providing an intrinsic curvature for three dimensional manifolds. In this article we explain why the naive finite difference method fails in general and provide explicit, semi-implicit and Newton solvers which perform better by enforcing a convexity type constraint needed for the ellipticity of the equation itself. We build a monotone wide stencil finite difference discretization, which is less accurate but provable convergent as a result of the Barles-Souganidis theory. Solutions with both discretizations are found 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.
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