SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 3, Page 1096-1118, January 2021.
This work introduces an extension of the classical local Fourier analysis (LFA) in which the discrete operator is described by a scalar stencil or stencils. First, we extend the scalar stencil to a matrix-stencil, whose coefficients are matrices rather than scalars and which is defined simply on a node-based infinite grid based on a recent work [Y. He, Numer. Linear Algebra Appl., to appear]. Meanwhile, we extend the symbols of stencil operators to matrix-stencil operators. Then, we prove that any scalar stencil operator, no matter how complicated it is, can also be described by a matrix-stencil operator. Furthermore, we prove that the symbols based on the scalar stencils and matrix-stencils of a given discrete operator are unitarily similar, i.e., the symbols have the same spectrum and norms. This connection allows us to develop a simple and unified framework of two-grid LFA based on matrix-stencils that are defined on node-based grids. This framework fits the finite element and difference discretizations very well. It results in a unified symbol computation for the discrete operator and the associated grid-transfer operators in a two-grid method. Finally, some discrete operators arising from finite element and difference discretizations are presented to illustrate the simplicity of this generalized LFA and its use.

中文翻译：

---

使用矩阵模板的局部傅立叶分析的通用和统一框架

SIAM Journal on Matrix Analysis and Applications，第 42 卷，第 3 期，第 1096-1118 页，2021 年 1 月。
这项工作引入了经典局部傅里叶分析 (LFA) 的扩展，其中离散算子由一个或多个标量模板描述。首先，我们将标量模板扩展为矩阵模板，其系数是矩阵而不是标量，并且基于最近的工作 [Y. 他，数字。线性代数应用程序，出现]。同时，我们将模板运算符的符号扩展到矩阵模板运算符。然后，我们证明了任何标量模板算子，无论它有多复杂，也可以用矩阵模板算子来描述。此外，我们证明了基于给定离散算子的标量模板和矩阵模板的符号是酉相似的，即符号具有相同的频谱和范数。这种连接使我们能够基于在基于节点的网格上定义的矩阵模板开发一个简单而统一的双网格 LFA 框架。该框架非常适合有限元和差分离散化。它导致在双网格方法中对离散算子和相关的网格传输算子进行统一的符号计算。最后，提出了一些由有限元和差分离散化产生的离散算子，以说明这种广义 LFA 及其使用的简单性。