In this paper, we use moving mesh finite element method based upon 4P1−
P1 element to solve the time-dependent Navier-Stokes equations in 2D. Two-layer
nested meshes are used including velocity mesh and pressure mesh, and velocity mesh
can be obtained by globally refining pressure mesh. We use hierarchy geometry tree
to store the nested meshes. This data structure make convienence for adaptive mesh
method and the construction of multigrid preconditioning. Several numerical problems
are used to show the effect of moving mesh.