Solving Navier-Stokes Equations with Stationary and Moving Interfaces on Unfitted Meshes

This paper introduces a high-order immersed finite element (IFE) method to solve two-phase incompressible Navier-Stokes equations on interface-unfitted meshes. In spatial discretization, we use the newly developed immersed P-2-P-1 Taylor-Hood finite element. The unisolvency of new IFE basis functions is theoretically established. We introduce an enhanced partially penalized IFE method which includes the penalization on both interface edges and the interface itself. Ghost penalties are also added for pressure robustness. In temporal discretization, theta-schemes and backward differentiation formulas are adopted. Newton's method is used to handle the nonlinear advection. The proposed method completely circumvent re-meshing in tackling moving-interface problems. Thanks to the isomorphism of our IFE spaces with the standard finite element spaces, the new method enables efficient updates of global matrices, which significantly reduces the overall computational cost. Comprehensive numerical experiments show that the proposed method is third-order convergent for velocity and second-order for pressure in both stationary and moving interface cases.