Moving Mesh Finite Element Methods for the Incompressible Navier-Stokes Equations

This work presents the first effort in designing a moving mesh algorithm to solve the incompressible Navier–Stokes equations in the primitive variables formulation. The main difficulty in developing this moving mesh scheme is how to keep it divergence-free for the velocity field at each time level. The proposed numerical scheme extends a recent moving grid method based on harmonic mapping [R. Li, T. Tang, and P. W. Zhang, J. Comput. Phys., 170 (2001), pp. 562–588], which decouples the PDE solver and the mesh-moving algorithm. This approach requires interpolating the solution on the newly generated mesh. Designing a divergence-free-preserving interpolation algorithm is the first goal of this work. Selecting suitable monitor functions is important and is found challenging for the incompressible flow simulations, which is the second goal of this study. The performance of the moving mesh scheme is tested on the standard periodic double shear layer problem. No spurious vorticity patterns appear when even fairly coarse grids are used.