Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

Abstract

We study discretizations by powers of discrete Laplacians of fully nonlinear equations. Our problems are parabolic and of order σ є (0,2) since they involve fractional Laplace operators (-Δ)σ/2. They arise e.g. in control and game theory as dynamic programming equations, and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations which are 2nd order accurate for all values of σ. The accuracy of previous approximations depend on σ and are worse when σ is close to 2. We show that the schemes are monotone, consistent, L∞-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We present several numerical examples. 2020 Mathematics Subject Classification. 49L25, 35J60, 34K37, 35R11, 35J70, 45K05, 49L25, 49M25, 93E20, 65N06, 65R20, 65N12.