In this work we study the finite difference method for the fractional diffusion equation with high-dimensional hyper-singular integral fractional Laplacian. We first propose a simple and easy-to-implement discrete approximation, i.e., fractional centered difference scheme with γth-order ($\gamma \le 2$) convergence for the fractional operator. Based on the established approximation, we then construct a finite difference scheme to solve fractional diffusion equations and analyze the stability and convergence in discrete energy norm ($0<\alpha \le 2$) and in discrete maximum norm ($1<\alpha \le 2$). We further present a fast solver for the linear system which is obtained by discretization on rectangular domain and use the fictitious domain method to extend the fast solver to the non-rectangular one. Several numerical results are provided to support our theoretical results.

在这项工作中，我们研究具有高维超奇异积分拉普拉斯算子的分数阶扩散方程的有限差分方法。我们首先提出一种简单且易于实现的离散逼近，即具有γ阶（$\gamma \le 2$）对小数运算符的收敛。在建立的近似值的基础上，我们构造一个有限差分方案来求解分数阶扩散方程，并分析离散能量范数的稳定性和收敛性（$0<\alpha \le 2$）和离散最大范数（$\mathrm{1个}<\alpha \le 2$）。我们进一步提出了一种用于线性系统的快速求解器，它是通过在矩形域上离散化而获得的，并使用虚拟域方法将快速求解器扩展到非矩形求解器。提供了一些数值结果来支持我们的理论结果。