A Nonconforming Nitsche’s Extended Finite Element Method for Stokes Interface Problems

A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems

“…This Nitsche's XFE method (NXFEM) was originally considered in [19] to solve the elliptic interface problems. Then a large number of related methods have been developed, such as [2,6,9,10,22,24,32,34,37,38] for elliptic interface problems, [3,11,20,25,35,36] for Stokes interface problems and [31] for Oseen problems.…”

“…We remark that the stability techniques are also used to solve Stokes problems with the interface in the context of fictitious domain method, where the extra penalty terms for the jumps in the normal velocity and pressure gradients near the interface were adopted to stabilize pressure in [30]. Very recently, the nonconforming-P 1 /P 0 NXFEM for a steady state Stokes interface problem was considered in [35]. The arithmetic averages were used and some stabilization terms were defined on interface edges and cut edges.…”

“…In this paper, we will use the nonconforming NXFEM of [22] and propose an accurate and stable extended finite element method for Stokes interface problems based on nonconforming-P 1 /P 0 shape functions with the unfitted-interface mesh. Although the same spaces considered in this paper (compared to [13,35]), we mention the following contributions of this paper. Instead of the weights involving the viscosity parameters and subareas in [13] and the arithmetic averages in [35], harmonic weight fluxes only involving the viscosity parameters are used on the interface.…”

A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces

“…Γ(t) If the interface does not change with respect to time, then conventional finite element methods [36] can solve parabolic interface problems satisfactorily provided that body-fitting meshes are used [2,4,6]. A body-fitting mesh has to be constructed according to the interface such that each element is essentially on one side of the interface and only touches the interface on its vertices, see the illustration in Figure 2.…”

“…Comparing with conventional fitted-mesh methods, such as classical FE and DG methods, the unfitted-mesh methods do not require the alignment of the mesh with a prescribed nontrivial interface; hence it is more desirable for time-dependent problems with moving interfaces. In the past decades, several unfitted-mesh methods have been developed for solving Stokes interface problems, such as CutFEM [15], Nitsche's FEM [36], XFEM [9], fictitious domain FEM [31,34], to name only a few. The immersed finite element method (IFEM) [24,26,18,11,14,30] is a class of unfitted-mesh finite element methods for solving interface problems.…”