A two-grid discretization method for nonlinear Schrödinger equation by mixed finite element methods.

In this paper, we investigate a two-grid discretization method for the two-dimensional time-dependent nonlinear Schrödinger equation by mixed finite element methods. Firstly, we solve original nonlinear Schrödinger equation on a much coarser grid. Then, we solve linear Schrödinger equation on the fine grid. We also propose the error estimate of the two-grid solution with the exact solution in L 2 -norm with order O (h k + 1 + H 2 k + 2). It is shown that our two-grid algorithm can achieve asymptotically optimal approximations as long as the mesh sizes satisfy h = O (H 2). Finally, two numerical experiments in the RT 0 space are provided to partly verify the accuracy and efficiency of the two-grid algorithm. [ABSTRACT FROM AUTHOR]