Numerical Methods for the 2-Hessian Elliptic Partial Differential Equation

2015/02/17 by Froese, Brittany D., Oberman, Adam M., Salvador, Tiago · 1 citation
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1502.04969

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.

Cited by

• Numerical methods for fully nonlinear degenerate diffusions

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