Research On Algebraic Multigrid Method And Application To Several Preconditioned Krylov Subsapce Methods

In this thesis the algebraic multigrid method is considered for solving system equations which arise from a finite element discretization of partial differential equations. First, we introduce the algebraic multigrid. Then the coarsening strategies for differentmeshes which we encounter in practice are presented in this paper. The algebraic mulitigrid is used as a preconditioner in several preconditioned Krylov subspace methods to solve the system equations.In this thesis, the Krylov subspace methods are introduced, such as BiConjugate Gradient method and Quasi-Minimal Residual method. Preconditioned BiConjugate Gradient method and preconditioned Quasi-Minimal Residual method are derived from the BiConjugate Gradient method and Quasi-Minimal Residual method.Finally, numerical studies are given for solving system equations which arise from a finite element dicretization of partial differential equations. The coarsened meshes are given in figures. Several solvers are applied and their convergence behaviors are compared. AMG preconditioned methods have higher convergence than other methods.