A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

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• 作者：Marco Discacciati ; Ricardo Oyarzúa
• 关键词：Mathematics Subject Classification65N15 ; 65N30 ; 76D05 ; 76S05
• 刊名：Numerische Mathematik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：135
• 期：2
• 页码：571-606
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Mathematics, general; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation; Appl.Mathematics/Comput
• 出版者：Springer Berlin Heidelberg
• ISSN：0945-3245
• 卷排序：135

文摘

In this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow. Flows are governed by the Navier–Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We consider the standard mixed formulation in the Navier–Stokes domain and the dual-mixed one in the Darcy region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported.