In this paper, we propose a new mixed finite element method for a swelling clay model with secondary consolidation, which is a process of the volume of saturated clay decreasing with time after the completion of primary consolidation. To better describe the multi-physics processes underlying in the original model, we firstly reformulate the swelling clay model with secondary consolidation by introducing a new variable to transform the fluid-solid coupling problem to a fluid coupled problem. Then we give the PDE analysis to prove the existence and uniqueness of the weak solution. And we design a fully discrete time-stepping scheme to use mixed finite element method with $P_2-P_1-P_1$ element pairs for the space variables and backward Euler method for the time variable, give the stability analysis and the optimal convergence order error estimates. Finally, we show some numerical examples to verify the theoretical results and significant effect on the displacement of the secondary consolidation term.

在本文中，我们针对具有二次固结的膨胀粘土模型提出了一种新的混合有限元方法，该模型是在一次固结完成后饱和粘土体积随时间减小的过程。为了更好地描述原始模型中的多物理过程，我们首先通过引入一个新变量将流固耦合问题转换为流体耦合问题来重新构建具有二次固结的膨胀粘土模型。然后我们给出 PDE 分析来证明弱解的存在性和唯一性。我们设计了一个完全离散的时间步长方案来使用混合有限元方法${磷}_2-{磷}_1-{磷}_1$空间变量的元素对和时间变量的后向欧拉法，给出了稳定性分析和最优收敛阶误差估计。最后，我们给出了一些数值例子来验证理论结果和对二次固结项位移的显着影响。