Modified two-grid method for solving coupled Navier-Stokes/Darcy model based on Newton iteration

Abstract

A new decoupled two-gird algorithm with the Newton iteration is proposed for solving the coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. Moreover the error estimate is given, which shows that the same order of accuracy can be achieved as solving the system directly in the fine mesh when h = H 2 . Both theoretical analysis and numerical experiments illustrate the efficiency of the algorithm for solving the coupled problem. §1 IntroductionThere is an increasing interest in studying coupling incompressible fluid flow and porous media flow. This complex phenomena can be found in many sciences such as geosciences and health sciences. For example, the problems of interaction of rivers with groundwater and the model of blood flow and organs have been researched in recent years. The fluid flow and the porous media flow are modeled by the Navier-Stokes equations and Darcy's law separately. The coupled models lead to various mathematical and numerical difficulties. First of all, the interface conditions of the coupled models involve different control variables from local models. Secondly, the local models may have complex, or even non-linear forms. Additionally, when we consider the coupled models, the cost of computation is enormously increasing compared with the each single problem.In general, coupled models can be solved in two kinds of approaches. One is direct method, and another is to decouple the coupled models and apply the corresponding local solver respectively. In past decades, there have been many direct solvers to be proposed. Chidyagwai and Rivière applied the continuous Galerkin (CG) and discontinuous Galerkin(DG) methods for the Navier-Stokes and Darcy models individually [9]. In [14], Girault and Rivière studied an Received: 2014-04-21. MR Subject Classification: 35B35, 65L15, 60G40.

Defect-deferred correction method for the non-stationary coupled Stokes/Darcy model

“…In addition, there always exists the mesh-independent constant C and the finite elements (u h , p f ) ∈ (U h , Q f h ), that satisfy the following optimal approximation property [27]:…”