Abstract
This paper is concerned with a novel methodology of smoothing analysis process of multicolor point relaxation by multigrid method for solving elliptically partial differential equations (PDEs). The objective was firstly focused on the two-color relaxation technique on the local Fourier analysis (LFA) and then generalized to the multicolor problem. As a key starting point of the problems under consideration, the mathematical constitutions among Fourier modes with various frequencies were constructed as a base to expand two-color to multicolor smoothing analyses. Two different invariant subspaces based on the 2h-harmonics for the two-color relaxation with two and four Fourier modes were constructed and successfully used in smoothing analysis process of Poisson’s equation for the two-color point Jacobi relaxation. Finally, the two-color smoothing analysis was generalized to the multicolor smoothing analysis problems by multigrid method based on the invariant subspaces constructed.
Highlights
Multigrid methods [1,2,3,4,5,6] are generally considered as one of the fastest numerical methods for solving complex partial differential equations (PDEs), for example, Navier-Stokes equation in computational fluid dynamics (CFD)
In order to obtain a Fourier representation of the mcolor point relaxation, let Shmc(ω) be the above complete mcolor point relaxation operator and let Shβ(ω) be the βth subrelaxation (β ∈ Λ m); the m-color point relaxation is expressed as m−1
The results are generalized to the m-color point relaxation and extended to a 3D system
Summary
Multigrid methods [1,2,3,4,5,6] are generally considered as one of the fastest numerical methods for solving complex partial differential equations (PDEs), for example, Navier-Stokes equation in computational fluid dynamics (CFD). The LFA monograph by Wienands and Joppich [11] provides an excellent background for experimenting with Fourier analysis. Recent advances in this context included LFA for triangular grids [13, 14], hexagonal meshes [15], semistructured meshes [16], multigrid with overlapping smoothers [17], multigrid with a preconditioner as parameters [18], and full multigrid method [19]. A novel smoothing analysis process of multicolor relaxation on LFA is provided with details. By the two invariant subspaces based on the 2h-harmonics the two-color smoothing analysis process is well generalized to the multicolor problems
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