On Stokes--Ritz projection and multistep backward differentiation schemes in decoupling the Stokes--Darcy model

Abstract

We analyze a parallel, noniterative, multiphysics domain decomposition method for decoupling the Stokes–Darcy model with multistep backward differentiation schemes for the time discretization and finite elements for the spatial discretization. Based on a rigorous analysis of the Ritz projection error shown in this article, we prove almost optimal L2 convergence of the numerical solution. In order to estimate the Ritz projection error on the interface, which plays a key role in the error analysis of the Stokes–Darcy problem, we derive L∞ error estimate of the Stokes–Ritz projection under the stress boundary condition for the first time in the literature. The k-step backward differentiation schemes, which are important to improve the accuracy in time discretization with unconditional stability, are analyzed in a general framework for any k ≤ 5. The unconditional stability and high accuracy of these schemes can allow relatively larger time step sizes for given accuracy requirements and hence save a significant amount of computational cost.