Two-grid Method For The Time-dependent Schr(?)dinger Equation

The Schrodinger equation,also known as the Schrodinger wave equation,is the most basic equation of quantum mechanics.It describes the law of the state of the microscopic particles changing with the time.The Schrodinger equation is also a basic assumption of quantum mechanics.It is usually not easy to get the exact solution of the Schrodinger equation in the actual complex system.Therefore,many experts and scholars both at home and abroad pay attention to the study of the numerical solution.Among the numerous numerical methods,the two-grid method is a very important and efficient method.In this paper,we mainly study two-grid finite element method and two-grid mixed finite element method for the time-dependent Schrodinger equation on the quasi-uniform triangular mesh.In chapter 3,the fully discrete finite element scheme is obtained by the finite element of order k in space with backward Euler method in time for the time-dependent linear Schrodinger equation.The error analysis of the finite element solution is conducted by the elliptical projection operator.At the same time,the two-grid algorithm is constructed with decouple equations,and related error analysis is carried out.Finally,numerical experiments also verify that the two-grid algorithm has the same error accuracy and higher computational efficiency than the standard finite element method.In chapter 4,the semi-discrete scheme and the semi-discrete decoupling two-grid algorithm are obtained by mixed finite element of RTk in space for the time-dependent linear Schrodinger equation.The error estimation of semi discrete mixed finite element solution and semi-discrete two-grid solution are obtained using the basic approximation property of the mixed finite element and the elliptic-mixed projection.Furthermore,the full-discrete mixed finite element scheme and the full discrete two-grid algorithm with decouple equations are established by back-ward Euler method in time.The theoretical analysis is carried out to obtain error estimations of the full-discrete mixed finite element solution and the full-discrete two-grid solution,respectively.Finally,numerical experiments verify that the two-grid algorithm has the same error accuracy as the standard mixed finite element method,and is more efficient.in time.In chapter 5,the semi-discrete mixed finite element scheme is presented by the RTk mixed finite element in space for the time-dependent nonlinear Schrodinger equation.In addition,the semi-discrete two-grid algorithm is proposed for solving linear problems on the fine grid.The error estimation of semi-discrete mixed finite element.solution and two-grid solution are achieved.Then,the full-discrete mixed finite element scheme and a linearized full-discrete two-grid algorithm are proposed by backward Euler method in time.The error estimates of the full-discrete mixed finite element solution and the full-discrete two-grid solution are obtained,respec-tively.Finally,numerical examples show that the two-grid algorithm has the same error accuracy as the standard mixed finite element method,and greatly improve the computational efficiency.