Analysis of a joint Stokes–Darcy Ritz-projection and multi-step BDF schemes for decoupling the unsteady Navier–Stokes–Darcy model

In this article, we analyze a semi-implicit noniterative decoupling method for the unsteady Navier–Stokes–Darcy system, based on multi-step backward differentiation schemes for the time discretization and finite elements for the spatial discretization. The results in the previous time steps are utilized to directly predict the interface information for decomposing the Navier–Stokes and Darcy sub-domains without any iteration at each time step. A semi-implicit scheme is proposed to linearize the nonlinear convection. In order to analyze the convergence of the finite-element solution of the proposed method, we rigorously prove the error estimate in the L² norm for the joint Stokes–Darcy Ritz-projection, notably without relying on the H² regularity assumption of the elliptic problem. The general k-step backward differentiation formulae (1≤ k≤ 5) are analyzed within a general framework utilizing multiplier techniques, and mathematical induction is employed to address the terms arising from nonlinear advection in the error estimate. Numerical examples are provided to verify the theoretical conclusions and illustrate the proposed method.