Convergent Finite Difference Methods for Fully Nonlinear Elliptic Equations in Three Dimensions

Brittany Froese Hamfeldt, Jacob Lesniewski

Research output: Contribution to journalArticlepeer-review

Abstract

We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully placed along the boundary at high resolution. We introduce and analyze a least-squares approach to building consistent, monotone approximations of second directional derivatives on these grids. We then show how to efficiently approximate functions of the eigenvalues of the Hessian through a multi-level discretization of orthogonal coordinate frames in R3. The resulting schemes are monotone and fit within many recently developed convergence frameworks for fully nonlinear elliptic equations including non-classical Dirichlet problems that admit discontinuous solutions, Monge–Ampère type equations in optimal transport, and eigenvalue problems involving nonlinear elliptic operators. Computational examples demonstrate the success of this method on a wide range of challenging examples.

Original languageEnglish (US)
Article number35
JournalJournal of Scientific Computing
Volume90
Issue number1
DOIs
StatePublished - Jan 2022

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Fully nonlinear elliptic equations
  • Generalized finite difference methods
  • Three dimensions
  • Viscosity solutions

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