Multi-Grid Method For Elliptic Partial Differential Equations

In many practical engineering and physics problem, partial differential equation is one of the most important mathematical model. For the partial differential equations with finite difference method and finite element method, the general practice is to solve the region meshing, forming a subdivision of discrete equations on a grid, and then using the iterative method to solve. The traditional finite difference method or the finite element method based on grid division has obvious shortcomings:to improve the accuracy of discrete solutions, need to be encrypted mesh, and this will cause the computation to increase rapidly, in order to overcome this shortcoming, MultiGrid method was introduced into people.This paper first introduces the basic principle of MultiGrid, as well as the V cycle MultiGrid implementation of algorithm. Then use the V MultiGrid cycle algorithm for one-dimensional, two-dimensional elliptic partial differential equations, Discuss the mesh nodes effect on MultiGrid convergence speed and precision. Obtain to keep fast convergence conditions, the calculation of the MultiGrid is only O(n), where n is the number of grid nodes. Visibly, the MultiGrid is a high accuracy for solving partial differential equations, method of high effective.In addition, the paper also illustrates the similarities and differences of MultiGrid finite difference method and the MultiGrid finite element method, the difference restriction between operator and continuation operator. to deepen understanding of the process of MultiGrid.