A Monotone Discretization for the Fractional Obstacle Problem

We propose a monotone discretization method for obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over a bounded Lipschitz domain. Our approach is motivated by the success of the monotone discretization of the fractional Laplacian [SIAM J. Numer. Anal. 60(6), pp. 3052-3077, 2022]. By exploiting the problem's unique structure, we establish the uniform boundedness, existence, and uniqueness of the numerical solutions. Moreover, we employ the policy iteration method to efficiently solve discrete nonlinear problems and prove its convergence after a finite number of iterations. The improved policy iteration, adapted to the regularity result, exhibits superior performance by modifying the discretization in different regions. Several numerical examples are provided to illustrate the effectiveness of our method.