In this paper, two-grid finite element method is applied to solve the nonlinear Schrödinger equation. On one hand, prior estimates of the numerical solution on the coarse grid and the introduced temporal equation are derived. Subsequently, taking advantage of the mathematical induction method and an identical equation of the inner product, result with order $O(h^2+\tau )$ in $L^2$-norm is deduced on the coarse grid and unconditional boundedness of the numerical solution is deduced as a result. Unconditional superclose result with order $O(h^2+\tau )$ in $H^1$-norm is arrived at based on the above results. On the other hand, on the fine grid, an Euler scheme is constructed based on the numerical solution of the coarse grid. To achieve unconditional superconvergent estimate of order $O(H^4+h^2+\tau )$ in $H^1$-norm, we give fine estimate of the nonlinear term with $H^2=O\left(h\right)$. At last, numerical example is introduced to confirm the theoretical analysis. Here, $H$ and $h$ are the spatial parameters on the coarse grid and the fine grid, respectively, and $\tau$ is the time step.

本文采用二网格有限元法求解非线性薛定谔方程。一方面，推导出粗网格上数值解的先验估计和引入的时间方程。随后，利用数学归纳法和一个相同的内积方程，得到有阶$哦(H^2+\tau )$ 在 ${升}^2$-norm 在粗网格上推导出来，结果推导出数值解的无条件有界性。带订单的无条件超级关闭结果$哦(H^2+\tau )$ 在 $H^1$-norm 是根据上述结果得出的。另一方面，在细网格上，基于粗网格的数值解构造欧拉格式。实现阶次的无条件超收敛估计$哦(H^4+H^2+\tau )$ 在 $H^1$-范数，我们给出非线性项的精细估计 $H^2=哦\left(H\right)$. 最后通过数值算例验证了理论分析。这里，$H$ 和 $H$ 分别是粗网格和细网格上的空间参数， $\tau$ 是时间步长。