A $P_2$-$P_1$ Partially Penalized Immersed Finite Element Method for Stokes Interface Problems

In this article, we develop a Taylor-Hood immersed finite element (IFE) method to
solve two-dimensional Stokes interface problems. The $P_2$-$P_1$ local IFE spaces are constructed
using the least-squares approximation on an enlarged fictitious element. The partially penalized
IFE method with ghost penalty is employed for solving Stoke interface problems. Penalty terms
are imposed on both interface edges and the actual interface curves. Ghost penalty terms are
enforced to enhance the stability of the numerical scheme, especially for the pressure approximation. Optimal convergences are observed in various numerical experiments with different interface
shapes and coefficient configurations. The effects of the ghost penalty and the fictitious element
are also examined through numerical experiments.