Second Order Unconditionally Stable and Convergent Linearized Scheme for a Fluid-Fluid Interaction Model

In this paper, a fully discrete ﬁnite element scheme with second-order temporal accuracy is proposed for a ﬂuid-ﬂuid interaction model, which consists of two Navier-Stokes equations coupled by a linear interface condition. The proposed fully discrete scheme is a combination of a mixed ﬁnite element approximation for spatial discretization, the second-order backward diﬀerentiation formula for temporal discretization, the second-order Gear’s extrapolation approach for the interface terms and extrapolated treatments in linearization for the nonlinear terms. Moreover, the unconditional stability is established by rigorous analysis and error estimate for the fully discrete scheme is also derived. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the accuracy and eﬃciency of the proposed scheme.