2021 UNC Greensboro PDE Conference.
Electron. J. Diff. Eqns., Conference 26 (2022), pp. 59-95.


Title: A narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs

Authors: Xiaobing Feng (The Univ. of Tennessee, Knoxville, TN, USA)
         Thomas Lewis (The Univ. of North Carolina, Greensboro, NC, USA)
         Kellie Ward (The Univ. of North Carolina, Greensboro, NC, USA)
 
Abstract: 
This article develops a unified general framework for designing convergent
finite difference and discontinuous Galerkin methods
for approximating viscosity and regular solutions of fully nonlinear second order PDEs.
Unlike the well-known monotone (finite difference) framework,
the proposed new framework allows for the use of narrow stencils and unstructured grids
which makes it possible to construct high order methods.
The general framework is based on the concepts of consistency and g-monotonicity
which are both defined in terms of various numerical derivative operators.
Specific methods that satisfy the framework are constructed using numerical moments.
Admissibility, stability, and convergence properties are proved,
and numerical experiments are provided along with some computer implementation details.

Published August 25, 2022.
Math Subject Classifications: 65N06, 65N12.
Key Words: Fully nonlinear PDEs; viscosity solutions;  Monge-Ampere equation;
  Hamilton-Jacobi-Bellman equation;  narrow-stencil; generalized monotonicity;
  g-monotonicity; numerical operators; numerical moment.