In this paper, a two-sided variable-coefficient space-fractional diffusion equation with fractional Neumann boundary condition is considered. To conquer the weak singularity caused by nonlocal space-fractional differential operators, a fractional block-centered finite difference (BCFD) method on general nonuniform grids is proposed. However, this discretization still results in an unstructured dense coefficient matrix with huge memory requirement and computational complexity. To address this issue, a fast version fractional BCFD algorithm by employing the well-known sum-of-exponentials (SOE) approximation technique is also proposed. Based upon the Krylov subspace iterative methods, fast matrix-vector multiplications of the resulting coefficient matrices with any vector are developed, in which they can be implemented in only $\mathcal{O}(MN_{exp})$ operations per iteration without losing any accuracy compared to the direct solvers, where $N_{exp}\ll M$ is the number of exponentials in the SOE approximation. Moreover, the coefficient matrices do not necessarily need to be generated explicitly, while they can be stored in $\mathcal{O}(MN_{exp})$ memory by only storing some coefficient vectors. Numerical experiments are provided to demonstrate the efficiency and accuracy of the method.

翻译：本文研究了一类带有分数阶Neumann边界条件的双侧变系数空间分数阶扩散方程。为克服非局部空间分数阶微分算子引起的弱奇异性，提出了一种适用于一般非均匀网格的分数阶块中心有限差分（BCFD）方法。然而，该离散格式仍会产生非结构化稠密系数矩阵，存在巨大的内存需求和计算复杂度。为解决此问题，进一步提出采用经典指数和（SOE）逼近技术的快速分数阶BCFD算法。基于Krylov子空间迭代方法，开发了所得系数矩阵与任意向量的快速矩阵-向量乘法运算。与直接求解器相比，该方法在每次迭代中仅需$\mathcal{O}(MN_{exp})$次运算即可实现，且不损失任何精度，其中$N_{exp}\ll M$为SOE逼近中的指数项数量。此外，系数矩阵无需显式生成，仅通过存储若干系数向量即可实现$\mathcal{O}(MN_{exp})$量级的内存存储。数值实验验证了该方法的效率与精度。