A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration

• Published in: Fractal and fractional Vol. 7; no. 8; p. 634
• Main Authors: Han, Rubing, Wu, Shuonan, Zhou, Hao
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-08-2023
• Subjects: Barriers; Boundary conditions; bounded Lipschitz domain; Dirichlet problem; Discretization; Finite element analysis; fractional Laplacian; Iterative methods; monotone discretization; Numerical stability; obstacle problem; Partial differential equations; policy iteration
• ISSN: 2504-3110; 2504-3110
• DOI: 10.3390/fractalfract7080634

In recent years, the fractional Laplacian has attracted the attention of many researchers, the corresponding fractional obstacle problems have been widely applied in mathematical finance, particle systems, and elastic theory. Furthermore, the monotonicity of the numerical scheme is beneficial for numerical stability. The purpose of this work is to introduce a monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. Through successful monotone discretization of the fractional Laplacian, the monotonicity is preserved for the fractional obstacle problem and the uniform boundedness, existence, and uniqueness of the numerical solutions of the fractional obstacle problems are proved. A policy iteration is adopted to solve the discrete nonlinear problems, and the convergence after finite iterations can be proved through the monotonicity of the scheme. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the efficacy of the proposed method.