Local and parallel finite element methods for the coupled Stokes/Darcy model
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Local and parallel finite element methods for the coupled Stokes/Darcy model Guangzhi Du1 · Liyun Zuo2 Received: 10 April 2020 / Accepted: 22 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, based on two-grid discretizations, two kinds of local and parallel finite element methods are proposed and investigated for the coupled Stokes/Darcy model. Following the idea presented in Xu and Zhou (Math. Comput. 69, 881–909 1999). a classical local and parallel finite element method is proposed and investigated. To derive global continuous approximations, a new local and parallel finite element method is devised by combining the partition of unity. We theoretically analyze the resulting formulations and derive optimal error estimates. Numerical experiments are reported to assess the theoretical results. Keywords Stokes/Darcy model · Parallel finite element method · Partition of unity · Numerical analysis Mathematics Subject Classification (2010) 35M30 · 65N12 · 65N30 · 76D05 · 76S05
1 Introduction The mixed Stokes/Darcy model for coupling fluid and porous media flows has been studied intensively due to its various applications, such as flows in fractured porous media and coupled ground water and surface flows. Up to now, a large number of numerical schemes have been proposed and investigated for this model [2–4, 13, 14, 16–18, 20, 23, 28, 34]. This model has higher fidelity than either the Stokes or Darcy Guangzhi Du
[email protected] Liyun Zuo [email protected] 1
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
2
School of Mathematical Sciences, University of Jinan, Jinan 250022, China
Numerical Algorithms
problem on their own. However, differences between the Stokes equations and the Darcy equations in physical and mathematical properties, such as type and scale, and the mixture of these two local models lead to various difficulties in numerically solving the coupled Stokes/Darcy model. To overcome the difficulty of the strong coupling of the two different models in different domains in numerical approximation, Mu and Xu [20] proposed a decoupled two-grid scheme, in which the coupled problem is solved on a coarse mesh and subsequently the coupled system is decoupled on a fine mesh by utilizing the coarse grid solution to approximate the interface conditions. Later on, based on this decoupled idea, some two/multilevel algorithms [1, 5, 35] and local and parallel finite element methods [6, 7, 36] were proposed. As we know, the local and parallel finite element methods were first proposed by Xu and Zhou for solving a class of elliptic problems [26, 27]. Then, these methods were generalized to the Stokes equations [11, 24], the Navier-Stokes equations [12, 25], the Navier-Stokes/Darcy model [6, 7, 36], and others [19, 29]. The key idea of these methods is that for a solution to PDEs, the low-frequency components can be approximated well by a relatively coarse grid and high-frequency components can be computed on a fine gri
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