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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
Organizers
Zhen Li
,
Xin Liang
,
Zhi Ting Ma
,
Hamid Mofidi
,
Li Wang
,
Fan Sheng Xiong
,
Shuo Yang
,
Wu Yue Yang
Speaker
Shuonan Wu
Time
Monday, October 14, 2024 3:00 PM - 4:00 PM
Venue
A3-1-101
Online
Zoom 928 682 9093
(BIMSA)
Abstract
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.