Methods Based On Multilevel Residual Space

Numerical method of PDE is the core of scientific computing, at the same time how to solve large-scale linear system is one of the most important things in the research of numerical method for PDE. There are many efficient methods to solve large-scale linear system, among these methods,multi-grid method is one of the most effective and scalable methods.To construct an efficient method for special problems is popular among expertise mathematician of scientific com-puting.In this paper,we introduce a new technology of solving large-scale linear systems.The paper is divided into three parts. We introduce the background of multi-grid method in the first chapter, and then use a simple model problem to introduce the basic idea of two-grid method. We also introduce how to construct and choose the interpolate operator , restriction operator and discrete operator on coarser grid. After completing these things, we introduce more general multi-grid method and their description, such as V-cycle, W-cycle and full multi-grid method.In chapter two,we first introduce the basic concept of multilevel residual space and the use Schmidt orthogonalization method to obtain a group of orthogonal base.Then we combine the multilevel residual space and projection method to construct a method called Multi-grid-Petrove-Galerkin method.Then we extend this method from the case which is that the matrix of system is SPD to more general case that is the matrix has inverse even if it is not SPD.The we expand the base of the space and obtain the expanded algorithm of this method.We also introduce the basic idea and concept of the multi-grid conjugate method and give another algorithm of partial orthogonalization at last.In chapter three,we apply the methods introduced in the last part to several PDE such as Poisson equation,convection-diffusion equation and equation with large jumping coefficients.Through these numerical example and comparison with other methods we will find that the methods based on multilevel residual space is more efficient for some special problems.