Two-Grid Crank-Nicolson Finite Volume Element Method for the Time-Dependent Schrödinger Equation (Advances in Applied Mathematics and Mechanics, 2022)
In this paper, we construct a Crank-Nicolson finite volume element schemeand a two-grid decoupling algorithm for solving the time-dependent Schrödinger equation. Combining the idea of two-grid discretization, the decoupling algorithm involvessolving a small coupling system on a coarse grid space and a decoupling system withtwo independent Poisson problems on a fine grid space, which can ensure the accuracywhile the size of coarse grid is much coarser than that of fine grid. We further providethe optimal error estimate of these two schemes rigorously by using elliptic projectionoperator. Finally, numerical simulations are provided to verify the correctness of thetheoretical analysis.