A Two-Grid Algorithm of the Finite Element Method for the Two-Dimensional Time-Dependent Schrödinger Equation

Abstract

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 H1-norm and carry out two numerical experiments.

Keywords

• two-grid algorithm
• Schrödinger equation
• finite element method
• backward Euler scheme
• Crank–Nicolson scheme