We consider a virtual element method in space for the nonlinear Ginzburg–Landau equation, while a linearized time-variable-step second order backward differentiation formula (BDF2) is adopted in time. The error splitting approach is used to prove the unconditional optimal error estimate of the derived scheme under the mild restriction on the ratio of adjacent time-steps ratios (similarly proposed in Liao and Zhang 2021, Zhang and Zhao 2021). By using the techniques of the discrete complementary convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels, we obtain the boundedness and error estimates of the solution of time-discrete system. Moreover, the optimal convergence in $L^2$-norm for the fully discrete scheme is finally derived. Numerical examples on a set of polygonal meshes are given to validate our theoretical results.

我们考虑非线性Ginzburg-Landau方程的空间虚元法，而在时间上采用线性化时变步长二阶反向微分公式（BDF2）。误差分割方法用于证明导出方案在对相邻时间步长比的比率有轻微限制的情况下的无条件最优误差估计（Liao and Zhang 2021、Zhang and Zhao 2021 中类似地提出）。通过使用离散互补卷积（DOC）核和离散互补卷积（DCC）核的技术，我们获得了时间离散系统解的有界性和误差估计。此外，最优收敛${\mathrm{大号}}^2$最终导出完全离散方案的 -norm。给出了一组多边形网格的数值例子来验证我们的理论结果。