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BIMSA Computational Math Seminar
Monotone Discretization of Integral Fractional Laplacian on Bounded Lipschitz Domains: Applications to the Fractional Obstacle Problem
Monotone Discretization of Integral Fractional Laplacian on Bounded Lipschitz Domains: Applications to the Fractional Obstacle Problem
演讲者
Shuonan Wu
时间
2024年10月14日 15:00 至 16:00
地点
A3-1-101
线上
Zoom 928 682 9093
(BIMSA)
摘要
We propose a monotone discretization method for the integral fractional Laplacian on bounded Lipschitz domains with homogeneous Dirichlet boundary conditions, specifically designed for solving fractional obstacle problems. Operating on unstructured grids in arbitrary dimensions, the method offers flexibility in approximating singular integrals over a domain that depends not only on the local grid size but also on the distance to the boundary, where the Hölder coefficient of the solution deteriorates. Using a discrete barrier function reflecting the distance to the boundary, we demonstrate optimal pointwise convergence rates in terms of the Hölder regularity of the data on quasi-uniform and graded grids.
Applying this monotone discretization to the (nonlinear) fractional obstacle problems, we establish the uniform boundedness, existence, and uniqueness of numerical solutions. Monotonicity naturally implies the convergence of the policy iteration. Subsequently, based on the nature of this problem, an improved policy iteration tailored to solution regularity is devised, exhibiting superior performance through adaptive discretization across diverse regions.
Applying this monotone discretization to the (nonlinear) fractional obstacle problems, we establish the uniform boundedness, existence, and uniqueness of numerical solutions. Monotonicity naturally implies the convergence of the policy iteration. Subsequently, based on the nature of this problem, an improved policy iteration tailored to solution regularity is devised, exhibiting superior performance through adaptive discretization across diverse regions.