Two-grid convergence theory for symmetric positive semidefinite linear systems

Xuefeng Xu
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Abstract

This paper is devoted to the convergence theory of two-grid methods for symmetric positive semidefinite linear systems, with particular focus on the singular case. In the case where the Moore--Penrose inverse of coarse-grid matrix is used as a coarse solver, we derive a succinct identity for characterizing the convergence factor of two-grid methods. More generally, we present some convergence estimates for two-grid methods with approximate coarse solvers, including both linear and general cases. A key feature of our analysis is that it does not require any additional assumptions on the system matrix, especially on its null space.
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对称正半有限线性系统的双网格收敛理论
本文主要研究对称正半有限线性系统的双网格方法的收敛理论,尤其关注正弦情况。在使用粗网格矩阵的 Moore-Penrose 逆作为粗求解器的情况下,我们推导出了描述双网格方法收敛因子的简明特性。更一般地说,我们提出了一些使用近似粗解器的双网格方法的收敛估计,包括线性和一般情况。我们分析的一个主要特点是,它不需要对系统矩阵,特别是其空空间做任何额外的假设。
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