Two-grid algorithm of the BDF2 finite element method for the two-dimensional linear Schrödinger equation

In this paper, we study the two-step backward differentiation formula (BDF2) finite element method for the two-dimensional time-dependent linear Schrödinger equation. Firstly, we obtain the BDF2 fully discrete finite element scheme of the Schrödinger equation, and analyze unconditional optimal error estimates by dividing the error analysis into temporal error and spatial error analysis, respectively. Secondly, we construct a two-grid algorithm of the BDF2 fully discrete finite element. With this method, the real and imaginary parts of the Schrödinger equation are decoupled, and the finite element solution on the fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two Poisson equations about real and imaginary parts on the fine grid. We also obtain the error estimate of the two-grid finite element solution with the exact solution in $H^1$ norm. Lastly, two numerical experiments are provided to verify theoretical analysis results.