In this paper, two implicit Backward Euler (BE) and Crank-Nicolson (CN) formulas of Ciarlet–Raviart mixed finite element method (FEM) are presented for the fourth order time-dependent singularly perturbed Bi-wave problem arising as a time-dependent version of Ginzburg-Landau-type model for $d$-wave superconductors by the bilinear element. The well-posedness of the weak solution and the approximation solutions of the considered problem are proved through Faedo-Galerkin technique and Brouwer fixed point theorem, respectively. The quasi-uniform and unconditional superconvergent estimates of $O(h^2+\tau )$ and $O(h^2+{\tau }^2)(h,$ the spatial parameter, and $\tau ,$ the time step) in the broken $H^1$- norm are obtained for the above formulas independent of the negative powers of the perturbation parameter $\delta$. Some numerical results are provided to illustrate our theoretical analysis.

本文针对四阶随时间变化的时间相关奇异摄动双波问题，提出了Ciarlet-Raviart混合有限元方法（FEM）的两个隐式Backward Euler（BE）和Crank-Nicolson（CN）公式。 Ginzburg-Landau型模型的从属版本 $d$双线性单元构成的超波。分别通过Faedo-Galerkin技术和Brouwer不动点定理证明了弱解的适定性和所考虑问题的逼近解。的准一致和无条件超收敛估计$Ø（H^2+\tau ）$ 和 $Ø（H^2+{\tau }^2）（H，$ 空间参数，以及 $\tau ，$ 时间步）在破碎 $H^{\mathrm{1个}}$-以上公式的范数与扰动参数的负幂无关 $\delta$。提供一些数值结果来说明我们的理论分析。