Local and parallel partition of unity scheme for the mixed Navier-Stokes-Darcy problem

In this paper, by combining the two-grid method with a partition of unity, a local and parallel partition of unity scheme is designed and investigated for the mixed Navier-Stokes-Darcy problem. The main features of the present method are the following: (1) Once a coarse solution is derived, both the linearized Navier-Stokes and Darcy subproblems with a finer grid are independent, which allows a parallel computing with little communication; (2) a partition of unity is considered to assemble the solutions obtained from the local subdomains to arrive at a global continuous approximation; (3) a further coarse grid correction is carried out to derive optimal error bounds for the fluid velocity and piezometric head in L2-norm. Moreover, the convergence of the proposed method is shown. Some numerical experiments are reported to demonstrate the theoretical results.