Two-grid methods with exact solution of the Galerkin coarse-grid system have been well studied by the multigrid community: an elegant identity has been established to characterize the convergence factor of exact two-grid methods. In practice, however, it is often too costly to solve the Galerkin coarse-grid system exactly, especially when its size is large. Instead, without essential loss of convergence speed, one may solve the coarse-grid system approximately. In this paper, we develop a new framework for analyzing the convergence of inexact two-grid methods: two-sided bounds for the energy norm of the error propagation matrix of inexact two-grid methods are presented. In the framework, a restricted smoother involved in the identity for exact two-grid convergence is used to measure how far the actual coarse-grid matrix deviates from the Galerkin one. As an application, we establish a unified convergence theory for multigrid methods.
翻译:多格网社区对精确解决Galerkin coarse- grid系统的双电网方法进行了很好的研究:为了确定精确的双电网方法的趋同因素的特征,已经确立了一种优雅的特性;然而,在实践中,精确地解决Galerkin coarse- grid系统往往费用太高,特别是在其大小很大的情况下。相反,在不基本丧失趋同速度的情况下,可以大致解决粗电网系统。在本文件中,我们开发了一个新的框架来分析不精确的双电网方法的趋同情况:提出了不精确的二电网方法传播矩阵错误的能量规范的双向界限。在框架中,使用精确的两电网聚合特性的有限光滑度来衡量实际的Galerkin矩阵与Galerkin 1号的距离。作为应用,我们为多格网方法建立统一的趋同理论。