A two-grid algorithm for the two-dimensional linear Schrödinger equation by the finite element method

In this paper, we construct a new two-grid algorithm in the two-step backward differentiation formula (BDF2) fully discrete finite element scheme for the two-dimensional linear Schrödinger equation. We first compute the solution of the Schrödinger equation on a much coarser grid with size H . Then we solve two elliptic equations about the real and imaginary parts on the fine grid with size h < < H . Consequently, our two-grid algorithm not only reduces CPU time, but also achieves the same accuracy of the numerical solution as the standard finite element method.