An Algebraic Multigrid Method On Anisotropic Quadrilateral Grids And A Two-grid Discretization Method For Coupled System Of Partial Differential Equations

This thesis includes two parts. In the first part, for a kind of isotropic elliptic boundary value problems, we are concerned with the joint adaptation of the anisotropic quadrilateral grids and the algebraic multigrid(AMG) solver. We find that the pure AMG methods are less efficient for some classical anisotropic quadrilateral grids. Furthermore, we design a new AMG method by an effective use of geometric information and analytic properties of underlying differential equation. The numerical results show that our AMG method is more robust and efficient.In the second part of the thesis, for a kind of coupled system of partial differential equations(PDEs), we first discuss two classes of AMG methods for the finite element equation, and then propose a new two-grid discretization method and iteration method to decouple the system. Finally, As a demonstration, we apply the above four methods to a Schrodinger equation. The numorical results indicate that the efficiency of the new two-grid discretization method and iteration method are better than the two classes of AMG methods.