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In this paper, we consider the effect of adding a coarse mesh correction to the two‐grid algorithm for the mixed Navier–Stokes/Darcy model. The method yields both L2 and H1 optimal velocity and piezometric head approximations and an L2 optimal pressure approximation. The method involves solving one small, coupled, nonlinear coarse mesh problem, two independent subproblems (linear Navier–Stokes equation and Darcy equation) on the fine mesh, and a correction problem on the coarse mesh. Theoretical analysis and numerical tests are done to indicate the significance of this method. [ABSTRACT FROM AUTHOR]

To investigate the fluid–fluid interaction problem, we study two first‐order in time, decoupled algorithms for a coupling system. This system consist of two heat equations, which are coupled by a condition that allows energy to pass back and forth across the interface. The first scheme is a combination of the three‐level implicit method with the coupling terms treated by the explicit leap‐frog method. Under a time step size condition, we prove that it is stable and first‐order convergent. To remove the time step size condition, we modify the treatment of the coupling interface term and propose a second scheme. Instead of making the interface term explicitly, we only lag the variable in the different subdomain onto the previous time step. As expected, the second scheme is proved to be stable and convergent unconditionally. Since both schemes only require the solution of two decoupled heat equations at each time step. Hence, they are efficient in computation and can be easily implemented by using legacy codes. The numerical tests also confirm our theoretical results. [ABSTRACT FROM AUTHOR]