通过GLT序列理论分析块雅可比/高斯-赛德尔及加法/乘法施瓦茨预条件子，并应用于区域分解离散化
Analysis of Block Jacobi/Gauss-Seidel and additive/multiplicative Schwarz preconditioning through the theory of GLT sequences, with applications to domain decomposition discretizations
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中文总结 AI 辅助
本文基于GLT序列理论定义BJ、BGS、AS、MS预条件子，分析其结构并证明相关符号性质，最后在等几何DDM中完成数值验证。
中文摘要 AI 辅助
当线性微分问题通过以网格精细度参数$n$为特征的线性数值方法进行离散化时，数值解的计算可归结为求解由矩阵$A_n$确定的线性离散问题，该矩阵的规模随$n$增大而增长。本文通过实例说明，离散化矩阵序列$\{A_n\}_n$通常属于广义局部托普利茨（GLT）序列类，即便该数值方法属于区域分解方法（DDM）家族亦是如此。针对DDM离散化矩阵，四种广泛使用的预条件子为块雅可比（BJ）、块高斯-赛德尔（BGS）、加法施瓦茨（AS）及乘法施瓦茨（MS）预条件子。本文针对任意多级块矩阵给出BJ、BGS、AS、MS预条件子的正式定义，这些定义及相关符号受GLT序列理论启发，旨在替代DDM领域常用的定义。我们分析将BJ、BGS、AS、MS预条件子应用于属于GLT序列$\{A_n\}_n$的多级块矩阵$A_n$时的结构。每个GLT序列$\{A_n\}_n$都唯一关联一个名为符号的特殊函数$\boldsymbol{\u03ba}$。我们证明，若$\boldsymbol{\u007bA_n\}_n$是符号为$\boldsymbol{\u03ba}$的GLT序列，则BJ、BGS及MS预条件子序列是符号为$\boldsymbol{\u03ba}$的GLT序列；对于AS预条件子，我们证明$\boldsymbol{\u007bP_n^{AS}(A_n)\}_n$是符号为$\boldsymbol{\u03ba^{AS}\approx\boldsymbol{\u03ba}}$的GLT序列，且当构造$\boldsymbol{\u0050_n^{AS}(A_n)}$所用子域的重叠度随$n\to\infty$趋于零时，$\boldsymbol{\u03ba^{AS}=\boldsymbol{\u03ba}}$。本文还在等几何DDM框架下对上述结果进行了数值验证。
英文摘要
When a linear differential problem is discretized by a linear numerical method characterized by a mesh fineness parameter $n$, the computation of the numerical solution reduces to solving a linear discrete problem identified by a matrix $A_n$ whose size grows with $n$. The sequence of discretization matrices $\{A_n\}_n$ often falls within the class of generalized locally Toeplitz (GLT) sequences, even when the numerical method belongs to the family of domain decomposition methods (DDMs), as illustrated herein through examples. Four widely used preconditioners for DDM discretization matrices are the block Jacobi (BJ), block Gauss--Seidel (BGS), additive Schwarz (AS), and multiplicative Schwarz (MS) preconditioners. In this paper, we provide formal definitions of the BJ/BGS/AS/MS preconditioners for arbitrary multilevel block matrices. These definitions and the associated notations are inspired by the theory of GLT sequences and are proposed as alternatives to those commonly used by the DDM community. We analyze the structure of the BJ/BGS/AS/MS preconditioners when applied to multilevel block matrices $A_n$ belonging to a GLT sequence $\{A_n\}_n$. Every GLT sequence $\{A_n\}_n$ is uniquely associated with a special function $κ$ called symbol. We prove that, if $\{A_n\}_n$ is a GLT sequence with symbol $κ$, then the sequences of the BJ, BGS, and MS preconditioners are GLT sequences with symbol $κ$. For the AS preconditioner, we prove that $\{P_n^{AS}(A_n)\}_n$ is a GLT sequence with symbol $κ^{AS}\approxκ$, and $κ^{AS}=κ$ whenever the overlaps in the subdomains used for the construction of $P_n^{AS}(A_n)$ vanish as $n\to\infty$. A numerical validation of these results in the context of isogeometric DDMs is presented.
