In this paper, a novel arbitrary Lagrangian–Eulerian (ALE) mapping, thus a novel ALE-mixed finite element method (FEM), is developed and analyzed for a type of mixed parabolic equations in a moving domain. By means of a specific stabilization technique, the mixed finite element of a stable Stokes-pair is utilized to discretize this problem on the ALE description, and, stability and a nearly optimal convergence results are obtained for both semi- and fully discrete ALE finite element approximations. Numerical experiments are carried out to validate all theoretical results. The developed novel ALE–FEM can be also similarly extended to a transient porous (Darcy’s) fluid flow problem in a moving domain as well as to Stokes/Darcy- or Stokes/Biot moving interface problem in the future.

在本文中，针对运动域中的一类混合抛物方程，开发了一种新颖的任意拉格朗日-欧拉（ALE）映射，从而一种新颖的ALE混合有限元方法（FEM）。通过特定的稳定技术，利用稳定Stokes对的混合有限元在ALE描述中离散化此问题，并且对于半离散和完全离散ALE有限元都获得了稳定性和接近最优的收敛结果近似值。进行数值实验以验证所有理论结果。所开发的新型ALE-FEM还可以类似地扩展到运动域中的瞬态多孔（Darcy's）流体流动问题，以及将来的Stokes / Darcy-或Stokes / Biot运动界面问题。