Meshfree finite difference approximations for functions of the eigenvalues of the Hessian

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Abstract

We introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the Hessian. Approximations are defined on unstructured point clouds, which allows for very complicated domains and a non-uniform distribution of discretisation points. The schemes are monotone, which ensures that they converge to the viscosity solution of the underlying PDE as long as the equation has a comparison principle. Numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds, complex domains, degenerate equations, and singular solutions.

Original languageEnglish (US)
Pages (from-to)75-99
Number of pages25
JournalNumerische Mathematik
Volume138
Issue number1
DOIs
StatePublished - Jan 1 2018

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics

Keywords

  • 35J15
  • 35J60
  • 35J70
  • 65N06
  • 65N12

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