Monotone two-scale methods for a class of integrodifferential operators and applications

TL;DR

This paper introduces a monotone, two-scale discretization method for a class of nonlocal integrodifferential operators, providing convergence rates and applications to free boundary problems and nonlinear nonlocal equations.

Contribution

It presents a novel monotone two-scale discretization scheme for nonlocal operators, with proven convergence and error estimates for related problems.

Findings

Established pointwise convergence rates for the discretization

Provided error estimates for free boundary problems

Developed a convergent scheme for nonlinear nonlocal equations

Abstract

We develop a monotone, two-scale discretization for a class of integrodifferential operators of order $2s$, $s \in (0,1)$. We apply it to develop numerical schemes, and derive pointwise convergence rates, for linear and obstacle problems governed by such operators. As applications of the monotonicity, we provide error estimates for free boundaries and a convergent numerical scheme for a concave fully nonlinear, nonlocal, problem.