Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

Constructive approximation on graded meshes for the integral fractional Laplacian

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian $(-Δ)^s$. We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is $C^s$ up to the boundary. We present the construction of graded bisection meshes by a greedy algorithm and derive quasi-optimal convergence rates for approximations to the solution of such a problem by continuous piecewise linear functions. The nonlinear Sobolev scale dictates the relation between regularity and approximability.

多重网格方法用于离散分数Sobolev空间

Multigrid Methods for Discrete Fractional Sobolev Spaces

本文提出了一种加性多重网格预处理方法用于正分数阶拉普拉斯算子，并证明了条件数的统一界。针对负分数阶情况，通过左乘右乘正分数阶预处理器与常规拉普拉斯算子来实现，同时详细讨论了实现中的问题并进行了数值实验验证。

分数阶障碍问题的一种单调离散化方法

A Monotone Discretization for the Fractional Obstacle Problem

本文针对分数阶障碍问题提出一种单调离散化方法，证明数值解的一致有界性、存在性与唯一性，并设计改进的策略迭代算法实现有限收敛，数值实验验证了方法的有效性。