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.

翻译：我们针对有界Lipschitz区域上满足齐次Dirichlet边界条件的积分分数阶Laplacian障碍问题，提出了一种单调离散化方法。该方法的提出受分数阶Laplacian单调离散化成功应用[SIAM J. Numer. Anal. 60(6), pp. 3052-3077, 2022]的启发。通过利用问题的独特结构，我们建立了数值解的一致有界性、存在性与唯一性。此外，我们采用策略迭代方法高效求解离散非线性问题，并证明了其在有限次迭代后的收敛性。基于正则性结果改进的策略迭代算法，通过在不同区域调整离散格式展现出更优性能。最后通过若干数值算例验证了本方法的有效性。