In this paper, we establish a reduced-order finite element (ROFE) method with very few degrees of freedom for the incompressible miscible displacement problem. Firstly, we construct the finite element (FE) method with second-order accuracy in time, where backward difference formulation is used for the time discretization and classical FE formulation is used for the space discretization. Optimal a priori error estimates for the FE solutions are proved. Secondly, we apply the proper orthogonal decomposition (POD) technique to develop the ROFE method, which can effectively reduce degrees of freedom and CPU time. Optimal a priori error estimates for the ROFE solutions are derived. Besides, we give the algorithm process of the ROFE method. Finally, some numerical examples are presented to verify the behavior of the ROFE method for piecewise linear element. And these examples imply the proposed method is feasible and effective for solving the incompressible miscible displacement problem.

在本文中，我们为不可压缩的混相位移问题建立了一个具有很少自由度的降阶有限元（ROFE）方法。首先，我们构造了具有二阶时间精度的有限元（FE）方法，其中时间离散化使用后向差分公式，空间离散化使用经典有限元公式。证明了有限元解决方案的最佳先验误差估计。其次，我们应用适当的正交分解（POD）技术来开发ROFE方法，可以有效地减少自由度和CPU时间。导出 ROFE 解决方案的最佳先验误差估计。此外，我们给出了ROFE方法的算法过程。最后，给出了一些数值例子来验证分段线性单元的 ROFE 方法的行为。这些例子意味着所提出的方法对于解决不可压缩的混相位移问题是可行和有效的。