Numerical methods for the 2-Hessian elliptic partial differential equation

Brittany D. Froese, Adam M. Oberman, Tiago Salvador

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 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.

Original languageEnglish (US)
Pages (from-to)209-236
Number of pages28
JournalIMA Journal of Numerical Analysis
Volume37
Issue number1
DOIs
StatePublished - Jan 1 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Hessian equation
  • ellipticity constraints
  • fully nonlinear elliptic partial differential equations
  • monotone schemes
  • nonlinear finite difference methods
  • viscosity solutions

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