Numerical analysis of a class of hemivariational inequalities governed by fluid–fluid coupled flow

We explore the well-posedness and conduct a numerical analysis of hemivariational inequalities for the coupled stationary Navier–Stokes/Navier–Stokes system. The interface condition involves the Clark subgradient and serves as a generalization of various interface interaction relations, including nonlinear transmission conditions and friction-type conditions. We present an existence and uniqueness result for a solution of the continuous model. We propose a domain decomposition approach to solve the coupled system and examine the convergence of iterations. Moreover, we use the finite element approximation to discretize the hemivariational inequality of the coupled system and derive error estimates, which lead to an optimal order for the $P1b$/$P1$ pair under appropriate solution regularity assumptions. Numerical results are reported that illustrate the optimal convergence order predicted by theoretical analysis.