SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 475-502, January 2021.
We derive optimal complex relaxation parameters minimizing smoothing factors associated with multigrid using red-black successive overrelaxation or damped Jacobi smoothing applied to a class of linear systems arising from discretized linear partial differential equations with a complex shift. Our analysis yields analytical formulas for smoothing factors as a function of the complex relaxation parameter, which may then be efficiently numerically minimized. Our results are applicable to second-order discretizations in arbitrary dimensions, and generalize earlier work of Irad Yavneh on optimal relaxation parameters in the real case. Our analysis is based on deriving a novel connection between the performance of successive overrelaxation as a smoother and as a solver, and is validated by numerical experiments on problems in two and three spatial dimensions, using both vertex- and cell-centered multigrid, with both constant and variable coefficients. In the variable coefficient case we assign different relaxation parameters to different grids points, which our framework allows us to do efficiently.

中文翻译：

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复位移线性系统多重网格中的最优复松弛参数

SIAM 矩阵分析与应用杂志，第 42 卷，第 2 期，第 475-502 页，2021 年 1 月。
我们使用红黑连续过松弛或阻尼雅可比平滑来最小化与多重网格相关的平滑因子，这些参数应用于一类线性系统，这些线性系统是由具有复杂移位的离散线性偏微分方程产生的。我们的分析产生了平滑因子的解析公式，作为复杂松弛参数的函数，然后可以有效地数值最小化。我们的结果适用于任意维度的二阶离散化，并概括了 Irad Yavneh 在实际情况下关于最佳松弛参数的早期工作。我们的分析基于推导出连续过度松弛作为平滑器和求解器的性能之间的新联系，并通过两个和三个空间维度问题的数值实验进行验证，使用以顶点和单元为中心的多重网格，具有常数和可变系数。在可变系数的情况下，我们为不同的网格点分配不同的松弛参数，我们的框架允许我们有效地做到这一点。