A Monotone Discretization for the Fractional Obstacle Problem

TL;DR

This paper presents a new monotone discretization method for the fractional obstacle problem, ensuring stability, existence, and uniqueness of solutions, with efficient policy iteration and numerical validation.

Contribution

It introduces a novel monotone discretization approach for fractional obstacle problems, improving solution stability and computational efficiency.

Findings

Established uniform boundedness and solution uniqueness.

Proposed an efficient policy iteration with finite convergence.

Numerical examples confirm the method's effectiveness.

Abstract

We introduce a novel monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. This problem is prevalent in mathematical finance, particle systems, and elastic theory. By leveraging insights from the successful monotone discretization of the fractional Laplacian, we establish uniform boundedness, solution existence, and uniqueness for the numerical solutions of the fractional obstacle problem. We employ a policy iteration approach for efficient solution of discrete nonlinear problems and prove its finite convergence. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the method's efficacy.