A Residual-Based A Posteriori Error Estimator for the Stokes-Darcy Coupled Problem
A Residual-Based A Posteriori Error Estimator for the Stokes-Darcy Coupled Problem
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DOI：
10.1137/080727646
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发表时间：
2010-04
期刊：
影响因子：
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通讯作者：
I. Babuska;G. Gatica
中科院分区：
文献类型：
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作者：
I. Babuska;G. Gatica
本文对流体与多孔介质耦合的一种新型混合有限元方法进行了后验误差分析。气流分别由Stokes方程和Darcy方程控制，传输条件由质量守恒、法向力平衡和beaver - joseph - saffman定律给出。有限元子空间考虑速度的Bernardi-Raugel和Raviart-Thomas单元，压力的分段常数，界面上定义的拉格朗日乘子的连续分段线性单元。针对这一耦合问题，我们给出了一个可靠、高效的基于残差的后验误差估计器。利用适当的辅助问题、所涉及的双线性形式所满足的多种连续补上条件、Clement插值算子和Raviart-Thomas算子的局部逼近性质，证明了该方法的可靠性。另一方面，Helmholtz分解、逆不等式以及基于三角形泡函数和边泡函数的局部化技术是证明估计器有效性的主要工具。只要稍加修改，我们的分析就可以推广到其他的有限元子空间，从而得到稳定的伽辽金格式。
In this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Stokes and Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The finite element subspaces consider Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, and local approximation properties of the Clement interpolant and Raviart-Thomas operator. On the other hand, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.