It is known that the solutions to space-fractional diffusion equations exhibit singularities near the boundary. Therefore, numerical methods discretized on the composite mesh, in which the mesh size is refined near the boundary, provide more precise approximations to the solutions. However, the coefficient matrices of the corresponding linear systems usually lose the diagonal dominance and are ill-conditioned, which in turn affect the convergence behavior of the iteration methods.In this work we study a finite volume method for two-sided fractional diffusion equations, in which a locally refined composite mesh is applied to capture the boundary singularities of the solutions. The diagonal blocks of the resulting three-by-three block linear system are proved to be positive-definite, based on which we propose an efficient block Gauss–Seidel method by decomposing the whole system into three subsystems with those diagonal blocks as the coefficient matrices. To further accelerate the convergence speed of the iteration, we use T. Chan's circulant preconditioner³¹ as the corresponding preconditioners and analyze the preconditioned matrices' spectra. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method and its strong potential in dealing with ill-conditioned problems. While we have not proved the convergence of the method in theory, the numerical experiments show that the proposed method is convergent.

中文翻译：

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复合网格上两侧分数阶扩散方程的一种有效的正定块预处理有限体积求解器

众所周知，空间分数扩散方程的解在边界附近表现出奇点。因此，在复合网格上离散化的数值方法，其中网格尺寸在边界附近细化，为解决方案提供更精确的近似值。然而，相应的线性系统的系数矩阵通常会失去对角线支配性和病态，这反过来又会影响迭代方法的收敛行为。 在这项工作中，我们研究了双边分数扩散方程的有限体积方法，其中应用局部改进的复合网格来捕获解的边界奇点。所得的三乘三块线性系统的对角块被证明是正定的，在此基础上，我们通过将整个系统分解为三个子系统，以这些对角块作为系数矩阵，提出了一种高效的块 Gauss-Seidel 方法。为了进一步加快迭代的收敛速度，我们使用了 T. Chan 的循环预处理器³¹作为相应的预处理器并分析预处理矩阵的光谱。数值实验证明了所提出方法的有效性和效率及其在处理病态问题方面的强大潜力。虽然我们没有在理论上证明该方法的收敛性，但数值实验表明该方法是收敛的。