Multilevel methods are among the most efficient numerical methods for solving large-scale linear systems that arise from discretized partial differential equations. The fundamental module of such methods is a two-level procedure, which consists of compatible relaxation and coarse-level correction. Regarding two-level convergence theory, most previous works focus on the case of exact (Galerkin) coarse solver. In practice, however, it is often too costly to solve the Galerkin coarse-level system exactly when its size is relatively large. Compared with the exact case, the convergence theory of inexact two-level methods is of more practical significance, while it is still less developed in the literature, especially when nonlinear coarse solvers are used. In this paper, we establish a general framework for analyzing the convergence of inexact two-level methods, in which the coarse-level system is solved approximately by an inner iterative procedure. The framework allows us to use various (linear, nonlinear, deterministic, randomized, or hybrid) solvers in the inner iterations, as long as the corresponding accuracy estimates are available.
翻译:多层次方法是解决来自离散部分差异方程式的大型线性系统的最有效的数字方法之一。这些方法的基本模块是一个两级程序,由兼容的放松和粗粗的校正组成。关于两级趋同理论,大多数以前的工作侧重于精确的(Galerkin)粗糙求解器。然而,在实践中,在Galerkin 粗皮级系统规模相对较大时,解决该系统往往费用太高。与确切的情况相比,不精确的两级方法的趋同理论具有更实际的意义,而在文献中,这种理论仍然不够发达,特别是在使用非线性粗略解析解析器时。在本文中,我们建立了一个分析不精确的两层次方法趋同的一般框架,在这一结构中,粗皮级系统大约通过一个内迭式程序解决。这个框架使我们能够在内部迭代法中使用各种解答器(线性、非线性、确定性、随机性、随机性或混合性),只要有相应的准确估计。