Non-symmetric Indefinite Second Order Elliptic Boundary Value Problems Multigrid Convergence

The thesis is devoted to discuss the V-cycle convergence of the multigrid methods for two order elliptic boundary value problem. It is well known that there are many efficient methods available for solving the discrete equations with approximate solutions of the symmetry positive define elliptic boundary value problem. To begin with, the smoothing effect of error which is decomposed into orthogonal vectors in grid function space is analyzed. The error-reducing effect in orthogonal subspace which is developed from high frequencies and low frequencies component has been analyzed. The convergence operator of the multigrid method for symmetry and positive define elliptic boundary value problem is proved to be less than 1. In the grid function space, the smoothing effect, smoothing degree and the convergence are discussed in orthogonal subspace which involve high frequencies and low frequencies respectively . Many papers present various analyses for comforming finite element multigrid methods which are often based on certain regularity and approximation assumptiom concerning smoother and the solution. In chapter three, we analyze the error convergence operator directly and analyze the V-cycle convergence for symmetry positive define elliptic boundary problem without regularity assumption. In chapter four, the coefficient which is not large for nonsymmetric and indefinite elliptic boundary value problems is considered. Base on the perturbation of symmetric and positive definite operator which present in paper[6], together with new assumption and lemma , the V-cycle convergence of the multigrid method for the nonsymmetric and indefinite elliptic boundary value problems is proved briefly. In chapter five, a few assumption and important theorem are developed, the multigrid arithmetic which use the Richardson iteration as its smoothing method can be founded. Finally the convergence of the multigrid method V-cycle for the nonsymmetric and indefinite elliptic problem without regularity assumption is analyzed in detail...