Multigrid Method For Immersed Finite Elements

Nowadays,multigrid methods have become important tools for solving partial differential equations.In this thesis,we discuss the relationship between the condition number of an immersed finite element(IFE)method and the jump of discontinuous coefficients as well as the location of the interface.We also propose multigrid methods for solving the algebraic system resulting from the IFE approximations of the high-contrast interface problem,and get the optimal convergence of the algorithms.Firstly,we give some preliminaries of the Sobolev space.Then we introduce the linear and bilinear immersed finite element spaces,and present error estimates of an IFE method.Secondly,we analyze the upper bound of the maximum eigenvalue and the lower bound of the minimum eigenvalue of the stiffness matrix.It is proved theoretically that the condition number is proportional to the jump,which is consistent with the result of the standard finite element.Moreover,the condition number is independent of the location of the interface.Numerical results are also given to verify our theoretical findings.Thirdly,we present multigrid methods for the one dimensional IFE discretization.Error estimates with respect to the L~2norm and weighted H~1semi-norm are proved to be optimal and independent of the jump.Then we show that the two-level immersed finite element spaces are nested.Further,the uniform convergence rate of multigrid methods may be obtained.Finally,we propose multigrid methods for the two dimensional IFE discretiza-tion.Error estimates of the rotated Q1 IFE method are given.Numerical experiments show that the rotated Q1 IFE method has optimal convergence rate independent of the jump of discontinuous coefficients.Based on the novel interface smoother,multigrid method is designed to yield robustness for large jump,which shows the efficiency of the algorithm.