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A Generalized and Unified Framework of Local Fourier Analysis Using Matrix-Stencils
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.
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