论文标题
不精确两网格方法的收敛分析：理论框架
Convergence analysis of inexact two-grid methods: A theoretical framework
论文作者
论文摘要
Multigrid是解决由离散的部分偏微分方程引起的大规模线性系统的最有效方法之一。作为多格里德分析的基础，两个网格理论在激励和分析跨部算法中起着重要作用。对于对称正定问题，具有精确溶液的Galerkin粗网格系统的两网格方法的收敛理论是成熟的，并且可以通过身份来表征精确的两个网格方法的收敛因子。与确切的情况相比，不精确的两网格方法的收敛理论（即，求解了粗网格系统大约求解）具有更实际的意义，而文献中它在文献中的发展仍然较低（原因是，不精确的粗网格校正的误差传播矩阵不是投影）。在本文中，我们开发了一个理论框架，用于对不精确方法的收敛分析。更具体地说，我们为不精确的两个网格方法的误差传播矩阵的能量标准提供了两侧边界，从中可以轻松地从中获得确切的两个网格收敛性的身份。作为一种应用，我们为多机方法建立了一个统一的收敛理论，该理论允许大约求解Coarsest网格系统。
Multigrid is one of the most efficient methods for solving large-scale linear systems that arise from discretized partial differential equations. As a foundation for multigrid analysis, two-grid theory plays an important role in motivating and analyzing multigrid algorithms. For symmetric positive definite problems, the convergence theory of two-grid methods with exact solution of the Galerkin coarse-grid system is mature, and the convergence factor of exact two-grid methods can be characterized by an identity. Compared with the exact case, the convergence theory of inexact two-grid methods (i.e., the coarse-grid system is solved approximately) is of more practical significance, while it is still less developed in the literature (one reason is that the error propagation matrix of inexact coarse-grid correction is not a projection). In this paper, we develop a theoretical framework for the convergence analysis of inexact two-grid methods. More specifically, we present two-sided bounds for the energy norm of the error propagation matrix of inexact two-grid methods, from which one can readily obtain the identity for exact two-grid convergence. As an application, we establish a unified convergence theory for multigrid methods, which allows the coarsest-grid system to be solved approximately.
