Optimal error estimates of both coupled and two-grid decoupled methods for a mixed Stokes–Stokes model

In this paper, we provide a coupled algorithm and a two-grid decoupled algorithm for a mixed Stokes–Stokes model, which is coupled by a nonlinear interface transmission condition. The coupled algorithm is to discretize the mixed model directly by standard finite element method. For the two-grid decoupled algorithm, we first solve the mixed model on a coarse grid, and update the solution on a fine grid by two separated Stokes problems. Under a hypothesis about the regularity of analytical solutions, optimal error estimates for two algorithms are achieved. Several numerical tests are given to verify our theoretical results.