In view of a class of two-dimensional time-dependent linear Schrodinger equation, the bilinear finite element method is used for a discretization in space in conjunction with backward Euler method used to obtain the full discrete finite element scheme in the temporal direction, thus constructing a two-grid algorithm for fully discrete finite element to decouple the real and imaginary parts of the coupled Schrodinger equation. Thus, the solution on fine grids is simplified as solving the original problem on coarse grids and solving two Poisson equations on fine grids. The numerical results show that the two-grid algorithm for fully discrete finite element is more efficient than the standard finite element method, and the numerical solution has the same optimal error order with the size of coarse and fine meshes meeting certain conditions.