Two-grid Methods For Solving Schr(o|")dinger-type Equation

The Schrodinger-type equation is a basic equation in quantum mechanics and is widely applied in the fields of science, technology and engineering etc. It has been acknowledged that the analytical solution of the Schrodinger-type equation is difficult to be found in practical researches due to its involving the complex func-tion and the nonlinear and the coupled system. Therefore the investigation of the numerical solutions of the Schrodinger-type equation is interested and necessary. Among all popular numerical approaches, the two-grid discretization method is an efficient and rapid one both in theoretical and practical researches since it is capable of decoupling and linearizing the systems.In this paper, we aim at investigating the two-grid finite element method and the two-grid finite volume element method for the Schrodinger-type equation. The main contents of this dissertation are as follows.The semi-discrete finite element methods for the time-dependent Schrodinger equation are first proposed and then proved to be convergent with optimal con-vergence orders. We also construct full-discrete two-grid finite element schemes by first discretizing the original problem on the coarse grid and then discretizing a decoupled system on the fine grid using the last time numerical solutions of the fine grid and the corresponding time numerical solutions of the coarse grid. Finally, we analyze the full-discrete finite element schemes and achieve the error estimate. Preliminary numerical results subsequently show the new-presented schemes are more effective than the standard finite element schemes.We also construct a full-discrete two-grid finite element scheme by discretizing a decoupled and linearizated system on the fine grid for the time-dependent nonlin-ear Schrodinger equation. The convergence of the proposed scheme is established and the error estimate is achieved. The listed numerical experiments indicate that the full-discrete scheme is more effective than the standard finite element scheme.We construct two-grid finite volume element schemes for the boundary-value problem of the time-independent Schrodinger equation to approximate the solution via the barycenter dual mesh. Then the schemes are proved to be convergent with optimal convergence orders. The given numerical experiments confirm the theoretical results and verify the effectiveness of the presented scheme.