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In this paper, we construct a new two-grid algorithm of the finite element method for the Schrödinger equation in backward Euler and Crank–Nicolson fully discrete schemes. On the coarser grid, we solve coupled real and imaginary parts of the Schrödinger equation. On the fine grid, real and imaginary parts of the Schrödinger equation are decoupled, and we solve the elliptic equation about real and imaginary parts, respectively. Then, we obtain error estimates of the exact solution with the two-grid solution in the H 1 -norm and carry out two numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study the semidiscrete mixed finite element scheme and construct a two‐grid algorithm for the two‐dimensional time‐dependent Schrödinger equation. We analyze error results of the mixed finite element solution in L2$$ {L}^2 $$‐norm by some projection operators. Then, we propose a two‐grid method of the semidiscrete mixed finite element. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two elliptic equations on the fine grid. We also obtain the error estimate of two‐grid solution with exact solution in L2$$ {L}^2 $$‐norm. Finally, a numerical experiment indicates that our two‐grid algorithm is more efficient than the standard mixed finite element method. [ABSTRACT FROM AUTHOR]

We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the L 2 -norm with order O (τ + h k + 1) . Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the L 2 -norm with order O (τ + h k + 1 + H k + 2) . The numerical experiment is provided to demonstrate the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]