Two-grid method for the two-dimensional time-dependent Schrödinger equation by the finite element method

Abstract

In this paper, we construct a backward Euler full-discrete two-grid finite element scheme for the two-dimensional time-dependent Schrödinger equation. With this method, the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two Poisson equations on the same fine grid. We analyze the error estimate of the standard finite element solution and the two-grid solution in the H1 norm. It is shown that the two-grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(hkk+1). Finally, a numerical experiment indicates that our two-grid algorithm is more efficient than the standard finite element method.