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Two-Grid Crank-Nicolson Finite Volume Element Method for the Time-Dependent Schrödinger Equation

. In this paper, we construct a Crank-Nicolson ﬁnite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schr¨odinger equation. Combining the idea of two-grid discretization, the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a ﬁne grid space, which can ensure the accuracy while the size of coarse grid is much coarser than that of ﬁne grid. We further provide the optimal error estimate of these two schemes rigorously by using elliptic projection operator. Finally, numerical simulations are provided to verify the correctness of the theoretical analysis.