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摘要
This paper aims to provide an insight on the mechanism for the fast convergence of multigrid (MG) V-cycle algorithms. With this goal, the Fourier transform analysis is implemented to compute quantitatively the frequency spectrum of components of the residual resulted from the MG iterations. MG method has been extensively employed to solve large-scale systems of algebraic equations. Hence, one needs to understand its fast convergence property and scalability mechanism before applying MG method to a system equation, which affects the computational efficiency directly. In a first step, a number of novel features of the MG algorithm are studied, including the relationships among the iteration numbers needed for converged solutions, condition number of the system matrix and the grid-spacing in this paper. In a second step, the complexity of the MG method is researched, and various numerical experiments have been conducted in solving algebraic equations of models in finite element method for both heat transfer problems and solid mechanics problems. It is found that if the number of iterations in different grids is fixed, the optimal iteration mode is determined according to the attenuation of the high-frequency and low-frequency components of the residual. Our results also show that the complexity of the MG method is close to O(N), meaning that it is a close-to-linear scalable solver, which echoes the findings given by other literature. This study has revealed an important insight that the linear scalability is achieved by mitigating the negative effects (stagnation in convergence) of the large condition number of the system matrix created on the finest grid, using a sequence of coarser grids with smaller and smaller condition numbers.
- 出版日期2019-6
- 单位太原理工大学
