A Stabilized Finite Element Method for the Stokes-Stokes Coupling Interface Problem

In this article, we present a stabilized finite element method to solve the Stokes-Stokes interface system. To couple the Stokes equations in two different domains, we utilize Nitsche's type interface conditions. Numerical instability arises due to the coupling conditions that produce the artificial energy transfer across the interface, which causes the local instability of the approximation of the pressure and velocity. In the present work, we propose a robust stabilized scheme by introducing a stabilization term and a consistency term to deal with the instability of the system, which also ensures the well-posedness of the algorithm. The continuity and weak coercivity are derived for the proposed stabilized scheme. The optimal convergence analysis is carried out rigorously. Finally, several numerical experiments are conducted to illustrate the applicability, validity, and efficiency of the numerical method for the Stokes-Stokes interface model.