J. Math. Fluid Mech. 20 (2018), 683–695
2017 Springer International Publishing AG
Journal of Mathematical
A Multi-grid Decoupling Method for the Coupled Fluid Flow with the Porous Media
Liyun Zuo and Guangzhi Du
Communicated by A. Quarteroni
Abstract. In this paper, we propose a multi-grid decoupling method for the coupled Navier–Stokes–Darcy problem with the
Beavers–Joseph–Saﬀman interface condition. The basic idea of the method is to ﬁrst solve a much smaller global problem
on a very coarse initial grid, then solve a linearized Newton problem and a Darcy problem in parallel on all the subsequently
reﬁned grids. Error bounds of the approximate solution for the proposed method are analyzed, and optimal error estimates
are obtained. Numerical experiments are conducted to verify the theoretical analysis and indicate the eﬀectiveness of the
Keywords. Navier–Stokes equations, Darcy’s law, decoupling, multi-grid technique, ﬁnite element method.
The coupling of ﬂuid ﬂow and porous media ﬂow has received more and more attention and has become
a very active research area in recent years. The major reason lies in its wide spectrum of real world
applications, including the environmental problem of groundwater contamination through rivers, the
industrial manufacturing of ﬁlters, the biological modeling of the coupled circulatory system with the
surrounding tissue, and so on.
The related numerical models are the coupled Stokes–Darcy model or the coupled Navier–Stokes–
Darcy model, especially the interface conditions include Beavers–Joseph–Saﬀman or Beavers–Joseph in-
terface conditions. Many workers pay attention to studying their mathematical analysis and numerical
methods, the readers can refer to [1,3–20,22–33,35–40]. However, most of previous works related to
the coupled Stokes–Darcy problem. In this paper, we focus on the nonlinear case, that is, the coupled
Navier–Stokes–Darcy problem. In , a decoupled and linearized two-grid algorithm was proposed and
investigated. To further improve the eﬀectiveness of solving the coupled Navier–Stokes–Darcy problem,
we now extend the algorithm in  and propose a multi-grid decoupling method. In this method, one
only ﬁrst solve a much smaller global problem on a very coarse initial grid, then solve a linearized Newton
problem and a Darcy problem in parallel on all the subsequently reﬁned grids. Moreover, we note that
the numerical analysis in  only obtained the optimal order of convergence for the porous media ﬂow
and half order lower than the optimal one for the ﬂuid ﬂow. In our work, we analyze the error bounds of
the approximate solution for the proposed method, and obtain the optimal error estimates for two ﬂows.
Numerical results well agree with the theoretical predictions, and also demonstrate the eﬀectiveness of
the proposed method.
The rest of the paper is organized as follows. In Sect. 2, the coupled Navier–Stokes–Darcy problem is
given. In Sect. 3, the multi-grid decoupling method is presented. In Sect. 4, convergence of the proposed
Subsidized by the National Nature Science Foundation of China (Grant Nos. 11701343, 11571274, 11401466) and the
Provincial Natural Science Foundation of Shandong (Grant No. ZR2017BA027).