Local Fourier analysis (LFA) serves a prominent role in the prediction of the convergence factor of multigrid methods for discretizations of PDEs. However, few discussions about the implementation of LFA for complicated discretizations of PDEs exist, such as higher order finite elements, as staggered meshes could lead to complex LFA representations of the grid-transfer operator. In this work, we prove that the LFA representation for d-dimensional PDEs is independent of the placement of degrees of freedom (DoFs). Intuitively speaking, the seeding of the unknowns has no direct effect on the LFA presentation, instead, it serves as a unitary transformation between different representations. Thus, different LFA representations for a given PDE have the same spectrum and norm. Furthermore, we provide a uniform representation in terms of the location of the unknowns, named simple representation, where we allocate all of the DoFs at nodes, resulting in a simple and unified way to compute the symbols of the discrete operators, especially for the grid-transfer operators. This simple representation can contribute to the generalization of the implementation of LFA for different types of discretizations and different problems, especially for higher order discretization methods.

中文翻译：

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局部傅立叶分析的放置独立性

局部傅里叶分析 (LFA) 在预测 PDE 离散化的多重网格方法的收敛因子方面起着重要作用。然而，关于对 PDE 的复杂离散化实施 LFA 的讨论很少，例如高阶有限元，因为交错网格可能导致网格传输算子的复杂 LFA 表示。在这项工作中，我们证明了d的 LFA 表示维 PDE 与自由度 (DoF) 的位置无关。直观地说，未知数的播种对 LFA 表示没有直接影响，相反，它充当不同表示之间的统一转换。因此，给定 PDE 的不同 LFA 表示具有相同的频谱和范数。此外，我们在未知数的位置方面提供了统一的表示，称为简单表示，我们在节点处分配所有的自由度，从而产生一种简单而统一的方式来计算离散算子的符号，特别是对于网格-转移运营商。这种简单的表示有助于泛化 LFA 对不同类型的离散化和不同问题的实现，尤其是对于高阶离散化方法。