The Multigrid V-Cycle Algorithms Based On The Finite Elements And Convergence Properties With Fourier Analysis

In the process of solving partial differential equations for fields of physics and engineering,the system equations established by the finite element method and grid-free method are large in scale,and most of them are sparse matrices.The direct method to solve the system equation requires high computer performance,and the computational work of applying the classical iterative method is not proportional to the number of unknown numbers of equations,which leads to low efficiency.Then the Multigrid method is widely concerned by researchers because of its fast convergence property and scalability mechanism.Therefore,the paper focus on the study of multigrid V-cycle algorithm based on finite element,and apply Fourier method to analyze the convergence performance of multigrid method.Firstly,a multigrid V-cycle algorithm based on finite element method is designed and implemented to solve one-dimensional,two-dimensional heat conduction problems and two-dimensional solid mechanics problems,respectively.From a large number of numerical examples,it is found that the multigrid V-cycle algorithms implemented in this paper can greatly improve the rate of solving the equations of the system.In order to research the mechanism of fast convergence of multigrid V-cycle algorithm,the paper first analyzes the reasons why multigrid method is superior to the traditional iterative method from the point of view of eigenvalue.Then a number of novel features of the multigrid method are studied,including the relationships among the iteration numbers needed for converged solutions,condition number of the system matrix and the grid-spacing.On the basis of the implementation of V-type multigrid method,the algorithm complexity analysis of multigrid method is also carried out.Taking the solution of Poisson equation on the rectangular region of the problem domain as an example,the Fourier method is used to analyze the reasons why the multigrid method is superior to the traditional iterative method,and then the Fourier transform is applied to the iterative error of the multigrid method for solving the heat conduction problem and the 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 multigrid 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.