A Uniformly Robust Staggered DG Method for the Unsteady Darcy-Forchheimer-Brinkman Problem

doi:10.1007/s42967-020-00106-1

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (1): 205-226.doi: 10.1007/s42967-020-00106-1

• ORIGINAL PAPERS • 上一篇下一篇

A Uniformly Robust Staggered DG Method for the Unsteady Darcy-Forchheimer-Brinkman Problem

Lina Zhao, Ming Fai Lam, Eric Chung

Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR, China

收稿日期:2020-08-25 修回日期:2020-11-17 出版日期:2022-03-20 发布日期:2022-03-01
通讯作者: Eric Chung, Lina Zhao, Ming Fai Lam E-mail:tschung@math.cuhk.edu.hk;lzhao@math.cuhk.edu.hk;mfam@math.cuhk.edu.hk
基金资助:
The research of Eric Chung is partially supported by the Hong Kong RGC General Research Fund (Project numbers 14304719 and 14302018) and CUHK Faculty of Science Direct Grant 2019-20.

A Uniformly Robust Staggered DG Method for the Unsteady Darcy-Forchheimer-Brinkman Problem

Lina Zhao, Ming Fai Lam, Eric Chung

Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR, China

Received:2020-08-25 Revised:2020-11-17 Online:2022-03-20 Published:2022-03-01
Contact: Eric Chung, Lina Zhao, Ming Fai Lam E-mail:tschung@math.cuhk.edu.hk;lzhao@math.cuhk.edu.hk;mfam@math.cuhk.edu.hk
Supported by:
The research of Eric Chung is partially supported by the Hong Kong RGC General Research Fund (Project numbers 14304719 and 14302018) and CUHK Faculty of Science Direct Grant 2019-20.

摘要/Abstract

摘要： In this paper, we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem. Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be fexibly applied to fairly general polygonal meshes. We relax the tangential continuity for velocity, which is the key ingredient in achieving the uniform robustness. We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme, and the theories indicate that the error estimates for velocity are independent of pressure. Several numerical experiments are presented to confrm the theoretical fndings.

关键词: Staggered DG method, Brinkman-Forchheimer, General meshes, Uniformly stable

Abstract: In this paper, we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem. Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be fexibly applied to fairly general polygonal meshes. We relax the tangential continuity for velocity, which is the key ingredient in achieving the uniform robustness. We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme, and the theories indicate that the error estimates for velocity are independent of pressure. Several numerical experiments are presented to confrm the theoretical fndings.

Key words: Staggered DG method, Brinkman-Forchheimer, General meshes, Uniformly stable

中图分类号:

Lina Zhao, Ming Fai Lam, Eric Chung. A Uniformly Robust Staggered DG Method for the Unsteady Darcy-Forchheimer-Brinkman Problem[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 205-226.

1. Amrouche, C., Girault, V.:Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czech. Math. J. 44, 109-140 (1994)
2. Badia, S., Codina, R.:Unifed stabilized fnite element formulations for the Stokes and the Darcy problems. SIAM J. Numer. Anal. 47, 1971-2000 (2009)
3. Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.:Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23, 199-214 (2013)
4. Burman, E., Hansbo, P.:Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem. Numer. Methods Partial Difer. Equ. 21, 986-997 (2005)
5. Burman, E., Hansbo, P.:A unifed stabilized method for Stokes' and Darcy's equations. J. Comput. Appl. Math. 198, 35-51 (2007)
6. Caocao, S., Yotov, I.:A Banach space mixed formulation for the unsteady Brinkman-Forchheimer equations. IMA J. Numer. Anal. (2020). https://doi.org/10.1093/imanum/draa035
7. Celebi, A.O., Kalantarov, V.K., Uğurlu, D.:On continuous dependence on coefcients of the Brinkman-Forchheimer equation. Appl. Math. Lett. 19, 801-807 (2006)
8. Chung, E.T., Ciarlet Jr., P., Yu, T.:Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell's equations on Cartesian grids. J. Comput. Phys. 235, 14-31 (2013)
9. Chung, E.T., Engquist, B.:Optimal discontinuous Galerkin methods for wave propagation. SIAM J. Numer. Anal. 44, 2131-2158 (2006)
10. Chung, E.T., Engquist, B.:Optimal discontinuous Galerkin methods for the acoustic wave equation in higher dimensions. SIAM J. Numer. Anal. 47, 3820-3848 (2009)
11. Chung, E.T., Park, E.-J., Zhao, L.:Guaranteed a posteriori error estimates for a staggered discontinuous Galerkin method. J. Sci. Comput. 75, 1079-1101 (2018)
12. Chung, E.T., Qiu, W.:Analysis of an SDG method for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 55, 543-569 (2017)
13. Ciarlet, P.G.:The Finite Element Method for Elliptic Problems. North-Holland Publishing, Amsterdam (1978)
14. Du, J., Chung, E.T.:An adaptive staggered discontinuous Galerkin method for the steady state convection-difusion equation. J. Sci. Comput. 77, 1490-1518 (2018)
15. Forchheimer, P.:Wasserbewegung durch Boden. Z. Ver. Deutsh. Ing. 45, 1782-1788 (1901)
16. Girault, V., Wheeler, M.F.:Numerical discretization of a Darcy-Forchheimer model. Numer. Math. 110, 161-198 (2008)
17. Guzmán, J., Neilan, M.:A family of nonconforming elements for the Brinkman problem. IMA J. Numer. Anal. 32, 1484-1508 (2012)
18. Kim, H.H., Chung, E.T., Lam, C.Y.:Mortar formulation for a class of staggered discontinuous Galerkin methods. Comput. Math. Appl. 71, 1568-1585 (2016)
19. Kim, H.H., Chung, E.T., Lee, C.S.:A staggered discontinuous Galerkin method for the Stokes system. SIAM J. Numer. Anal. 51, 3327-3350 (2013)
20. Kim, M.-Y., Park, E.-J.:Fully discrete mixed fnite element approximations for non-Darcy fows in porous media. Comput. Math. Appl. 38, 113-129 (1999)
21. Kim, D., Zhao, L., Park, E.-J.:Staggered DG methods for the pseudostress-velocity formulation of the Stokes equations on general meshes. SIAM J. Sci. Comput. 42, A2537-A2560 (2020)
22. Könnö, J., Stenberg, R.:H(div)-conforming fnite elements for the Brinkman problem. Math. Models Methods Appl. Sci. 21, 2227-2248 (2011)
23. Louaked, M., Seloula, N., Trabelsi, S.:Approximation of the unsteady Brinkman-Forchheimer equations by the pressure stabilization method. Numer. Methods Partial Difer. Equ. 33, 1949-1965 (2017)
24. Mardal, K.A., Tai, X.-C., Winther, R.:A robust fnite element method for Darcy-Stokes fow. SIAM J. Numer. Anal. 40, 1605-1631 (2002)
25. Pan, H., Rui, H.:Mixed element method for two-dimensional Darcy-Forchheimer model. J. Sci. Comput. 52, 563-587 (2012)
26. Park, E.-J.:Mixed fnite element methods for generalized Forchheimer fow in porous media. Numer. Methods Partial Difer. Equ. 21, 213-228 (2005)
27. Paye, L.E., Straughan, B.:Convergence and continuous dependence for the Brinkman-Forchheimer equations. Stud. Appl. Math. 102, 419-439 (1999)
28. Rui, H., Liu, W.:A two-grid block-centered fnite diference method for Darcy-Forchheimer fow in porous media. SIAM J. Numer. Anal. 53, 1941-1962 (2015)
29. Rui, H., Pan, H.:A block-centered fnite diference method for the Darcy-Forchheimer model. SIAM J. Numer. Anal. 50, 2612-2631 (2012)
30. Showalter, R.E.:Monotone Operators in Banach Spaces and Nonlinear Partial Diferential Equations. In Mathematics Surveys and Monographs, vol. 49. AMS, Providence (1997)
31. Tian, L., Guo, H., Jia, R., Yang, Y.:An h-adaptive local discontinuous Galerkin method for simulating wormhole propagation with Darcy-Forcheiner model. J. Sci. Comput. (2020). https://doi.org/10.1007/s10915-020-01135-x
32. Zhao, L., Chung, E.T., Lam, M.:A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits. Comput. Methods Appl. Mech. Engrg. (2020). https://doi.org/10.1016/j.cma.[2020]112986
33. Zhao, L., Chung, E. T., Park, E.-J., Zhou, G.:Staggered DG method for coupling of the Stokes and Darcy-Forchheimer problems. SIAM J. Numer. Anal. 29, 1-31 (2021)
34. Zhao, L., Park, E.-J.:A staggered discontinuous Galerkin method of minimal dimension on quadrilateral and polygonal meshes. SIAM J. Sci. Comput. 40, A2543-A2567 (2018)
35. Zhao, L., Park, E.-J.:A new hybrid staggered discontinuous Galerkin method on general meshes. J. Sci. Comput. (2020). https://doi.org/10.1007/s10915-019-01119-6
36. Zhao, L., Park, E.-J., Shin, D.-W.:A staggered DG method of minimal dimension for the Stokes equations on general meshes. Comput. Methods Appl. Mech. Energy 345, 854-875 (2019)
37. Zhao, L., Park, E.-J.:A lowest-order staggered DG method for the coupled Stokes-Darcy problem. IMA J. Numer. Anal. 40, 2871-2897 (2020)
38. Zhao, L., Park, E.-J.:A staggered cell-centered DG method for linear elasticity on polygonal meshes. SIAM J. Sci. Comput. 42, A2158-A2181 (2020)

[1]	Fei Li, Sha Li, Liuqiang Zhong, Wanwan Zhu. Modified Adaptive Two-Grid FEMs for Nonsymmetric or Indefinite Elliptic Problems[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 831-850.
[2]	Hongying Man, Shangyou Zhang. On the Superconvergence of a Conforming Mixed Finite Element for Linear Elasticity on Uniform n-Square Grids[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 851-867.
[3]	Jinhua Feng, Yanjun Li, Hai Bi, Yidu Yang. An Adaptive Conforming Mixed Method for the Modified Transmission Eigenvalue Problem for Absorbing Medium[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 1095-1114.
[4]	Xiu Ye, Shangyou Zhang. Two-Order Superconvergent CDG Finite Element Method for the Heat Equation on Triangular and Tetrahedral Meshes[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 411-426.
[5]	Kamana Porwal, Tanvi Wadhawan. Convergence Analysis of Non-conforming Finite Element Method for a Quasi-static Contact Problem[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 578-604.
[6]	Mengjiao Pan, Tatyana Sorokina, Shangyou Zhang. Interpolated Galerkin Finite Elements on Rectangular and Cuboid Meshes for the Biharmonic Equation[J]. Communications on Applied Mathematics and Computation, 2026, 8(2): 765-777.
[7]	Pratibha Shakya, Kamana Porwal. Adaptive Finite Element Method for an Elliptic Optimal Control Problem with Integral State Constraints[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 1-40.
[8]	Masaru Miyashita, Norikazu Saito. A Finite Element Method to a Periodic Steady-State Problem for an Electromagnetic Field System Using the Space-Time Finite-Element Exterior Calculus[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 41-62.
[9]	Jiachang Sun. GPA: Intrinsic Parallel Solver for the Discrete PDE Eigen-Problem[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 970-986.
[10]	Jun Hu, Xueqin Yang. Two Classes of Mixed Finite Element Methods for the Reissner-Mindlin Plate Problem[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 1098-1121.
[11]	Gang Bao, Peijun Li, Xiaokai Yuan. Convergence of the PML Method for the Biharmonic Wave Scattering Problem in Periodic Structures[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 1122-1145.
[12]	Xiu Ye, Shangyou Zhang. Constructing a CDG Finite Element with Order Two Superconvergence on Rectangular Meshes[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 411-425.
[13]	Shangyou Zhang. Conforming P₃ Divergence-Free Finite Elements for the Stokes Equations on Subquadrilateral Triangular Meshes[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 426-441.
[14]	Shangyou Zhang, Zhimin Zhang. A Locking-Free and Reduction-Free Conforming Finite Element Method for the Reissner-Mindlin Plate on Rectangular Meshes[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 470-484.
[15]	Qigang Liang, Wei Wang, Xuejun Xu. A Domain Decomposition Method for Nonconforming Finite Element Approximations of Eigenvalue Problems[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 606-636.

A Uniformly Robust Staggered DG Method for the Unsteady Darcy-Forchheimer-Brinkman Problem

A Uniformly Robust Staggered DG Method for the Unsteady Darcy-Forchheimer-Brinkman Problem

PDF

可视化

摘要/Abstract

引用本文

使用本文

相关文章 15

编辑推荐

Metrics

本文评价