We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space. The use of Crank-Nicolson in time and finite differences in space leads to dense Toeplitz-like linear systems. Multigrid strategies that exploit such structure are particularly effective when the fractional orders are both close to 2. We seek to investigate how structure-based multigrid approaches can be efficiently extended to the case where only one of the two fractional orders is close to 2, i.e., when the fractional equation shows an intrinsic anisotropy. Precisely, we design a multigrid (block-banded–banded-block) preconditioner whose grid transfer operator is obtained with a semi-coarsening technique and that has relaxed Jacobi as smoother. The Jacobi relaxation parameter is estimated by using an automatic symbol-based procedure. A further improvement in the robustness of the proposed multigrid method is attained using the V-cycle with semi-coarsening as smoother inside an outer full-coarsening. Several numerical results confirm that the resulting multigrid preconditioner is computationally effective and outperforms current state of the art techniques.

中文翻译：

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各向异性空间分数扩散方程的多重网格预处理器

我们专注于在空间中具有分数导数的二维时空扩散方程。在时间上和空间上的有限差异中使用Crank-Nicolson会导致密集的Toeplitz式线性系统。当分数阶都接近2时，利用这种结构的多重网格策略特别有效。我们试图研究基于结构的多重网格方法如何有效地扩展到两个分数阶中只有一个接近2的情况，即，当分数方程式显示本征各向异性时。精确地，我们设计了一个多网格（块带状带状块）预处理器，其预处理器通过半粗化技术获得，并且使Jacobi变得更平滑。Jacobi松弛参数通过使用基于符号的自动过程进行估算。使用带有半粗化效果的V循环在外部全粗化效果内更平滑，可以进一步提高所提出的多重网格方法的鲁棒性。几个数值结果证实了所得的多重网格预处理器在计算上是有效的，并且优于当前的技术水平。