Study On Superconvergence Of Finite Element Method For The Schrodinger Equation

The Schrodinger equation is the basic equation of quantum mechanics. It is widely used in nonlinear optics, plasma physics, electromagnetic wave theory, nuclear physics, quantum chemistry and so on. The Schrodinger equation is also a basic assumption of quantum mechanics, it can not be proved from what is more fundamental presumption and its correctness can only be verified by practice. In addition, it is not easy to obtain the exact solution of the Schrodinger equation which is very complex in reality. Therefore, there are more and more attentions and concerns about the study on the numerical solution.In this paper, the finite element super convergence results are researched on uniform rectangular meshes or generalized rectangular meshes for three types of Schrodinger equations.In Chapter 3, the semi-discrete scheme is obtained by rectangular Lagrange type finite element of order p in space for the two-dimensional time-dependent Schrodinger equation. The error analysis is conducted by the finite element in-terpolation error theory and the elliptic projection operator, and the supercon-vergence result is obtained between the semi-discrete numerical solution and the interpolation function of the exact solution, respectively. Furthermore, the global superconvergence is obtained by constructing the interpolation post-processing op-erator. Secondly, a fully discrete scheme is obtained by Crank-Nicolson method in time, and the superconvergence result is proved between the fully-discrete numeri-cal solution and the interpolation function of the exact solution. Finally, numerical examples with the order p= 1 are provided to verify theoretical results.In Chapter 4, it is discretized by the bilinear rectangular finite element for the two-dimensional time-independent nonlinear Schrodinger equation. The error analysis of the numerical solution is conducted by the elliptic projection operator, and the superconvergence result is obtained. In addition, the global superconver-gence is samely obtained by the interpolation post-processing technique. At last, numerical examples are provided to verify theoretical results.In Chapter 5, the semi-discrete scheme is firstly obtained by the bilinear rect-angular finite element in space for the two-dimensional time-dependent nonlinearSchrodinger equation. The superconvergence result between the semi-discrete nu-merical solution and the interpolation function of the exact solution is proved by the elliptic projection operator. In addition, the global superconvergence is ob-tained by the interpolation post-processing technique. Secondly, two fully-discrete schemes are obtained by backward Euler method and Crank-Nicolson method in time. The optimal order error estimates are proved in L2 norm between the nu-merical solution and the exact solution in both fully-discrete schemes. Finally, numerical examples are provided to verify theoretical results.