In this work, a local Fourier analysis is presented to study the convergence of multigrid methods based on additive Schwarz smoothers. This analysis is presented as a general framework which allows us to study these smoothers for any type of discretization and problem. The presented framework is crucial in practice since it allows one to know a priori the answer to questions such as what is the size of the patch to use within these relaxations, the size of the overlapping, or even the optimal values for the weights involved in the smoother. Results are shown for a class of additive and restricted additive Schwarz relaxations used within a multigrid framework applied to high-order finite-element discretizations and saddle point problems, which are two of the contexts in which these type of relaxations are widely used.

翻译：本文提出了一种局部傅里叶分析方法，用于研究基于加法Schwarz平滑子的多重网格方法的收敛性。该分析作为一个通用框架，使我们能够研究这类平滑子在任何离散化类型和问题中的表现。该框架具有重要实践意义，因为它可以预先回答如下问题：在松弛过程中应使用的补丁尺寸、重叠区域的大小，甚至平滑子中权重的优化取值。本文展示了将一类加性和限制性加法Schwarz松弛方法应用于高阶有限元离散化和鞍点问题时的结果——这两类问题正是此类松弛方法广泛应用的典型场景。