An unconditionally energy stable finite element scheme for a nonlinear fluid–fluid interaction model

Abstract

Abstract In this paper, we design a decoupled scheme for solving a fluid–fluid interaction problem, which includes two Navier–Stokes equations coupled by some nonlinear interface conditions. Compared with two decoupled schemes proposed by Connors et al. (2012, Decoupled time stepping methods for fluid–fluid interaction. SIAM J. Numer. Anal., 50, 1297–1319) for the fluid–fluid interaction problem, we deal with these nonlinear interface conditions by applying explicit scheme. The presented fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization and explicit treatment for the interface conditions and the nonlinear terms. Moreover, the unconditional energy stability is established and error estimate for the fully discrete scheme is also showed. Finally, some numerical experiments are provided to verify the theoretical results, which illustrate the accuracy and efficiency of the presented scheme.