A Discontinuous Galerkin Method for the Coupled Stokes and Darcy Problem
- PDF / 1,582,027 Bytes
- 27 Pages / 439.37 x 666.142 pts Page_size
- 91 Downloads / 183 Views
A Discontinuous Galerkin Method for the Coupled Stokes and Darcy Problem Jing Wen1 · Jian Su1 · Yinnian He1 · Hongbin Chen1 Received: 24 December 2019 / Revised: 5 September 2020 / Accepted: 8 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Combining the mixed discontinuous Galerkin method for the Darcy flow and the interior penalty discontinuous Galerkin methods for the Stokes problem, a locally conservative discrete scheme is proposed for numerically solving the coupled Stokes and Darcy problem. We prove the well-posedness of the solution of the proposed numerical scheme by boundedness, K-ellipticity and a discrete inf-sup condition. A priori error estimates, in proper norms are derived, and to verify the theoretical analysis, some numerical experiments are given. Keywords Stokes and Darcy problem · Discontinuous Galerkin methods · Priori error estimates
1 Introduction The coupled Stokes and Darcy model describes the interaction between free flow and porous media flow. Such systems arise, for example, in modeling the groundwater (aquifer) contamination through filtration and streams, and numerical modeling of this complicated interaction is a challenging work in both theoretical analysis and practical engineering applications. There are some related works of the coupled system. Based on the Beavers–Joseph–Saffman interface conditions [14] Layton, Schieweck, and Yotov [27] prove the existence and uniqueness of a weak solution of the coupled system and, present and analyze its numerical scheme by adopting continuous finite element methods to discretize the Stokes problem and mixed finite element methods (MFE) to discrete the Darcy problem. Rivière et al. [3,4,13] propose and analyze a locally conservative discrete scheme by employing discontinuous Galerkin (DG)
B
Hongbin Chen [email protected] Jing Wen [email protected] Jian Su [email protected] Yinnian He [email protected]
1
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China 0123456789().: V,-vol
123
26
Page 2 of 27
Journal of Scientific Computing
(2020) 85:26
methods and mixed finite element methods for the coupled Stokes and Darcy equations, and by utilizing the DG methods for the coupled Navier–Stokes and Darcy problem. In addition, based on DG methods and mixed finite element methods, a strongly conservative numerical scheme is given in [15] by this group. Fu and Lehrenfeld [16] propose a strongly conservative numerical scheme for the coupled system by considering hybrid discontinuous Galerkin methods (HDG) and mixed finite element methods. Based on a continuous trace approximation of velocity and a discontinuous trace approximation of pressure, Cesmelioglu et al. present a embedded-hybridized discontinuous Galerkin (EDG-HDG) finite element method [1] with strong mass conservation for the coupled Stokes–Darcy problem. Mixed discontinuous Galerkin (MDG) method [12] and discontinuous Galerkin (DG) methods [2,5,6,8,11] are two kinds of locally mass conservative num
Data Loading...
