Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

UNCONDITIONALLY STABLE GAUGE-UZAWA FINITE ELEMENT METHODS FOR THE DARCY-BRINKMAN EQUATIONS DRIVEN BY TEMPERATURE AND SALT CONCENTRATION

  • Yangwei Liao (College of Mathematics and System Science Xinjiang University) ;
  • Demin Liu (College of Mathematics and System Science Xinjiang University)
  • Received : 2023.01.31
  • Accepted : 2023.03.30
  • Published : 2024.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the Gauge-Uzawa methods for the Darcy-Brinkman equations driven by temperature and salt concentration (DBTC) are proposed. The first order backward difference formula is adopted to approximate the time derivative term, and the linear term is treated implicitly, the nonlinear terms are treated semi-implicit. In each time step, the coupling elliptic problems of velocity, temperature and salt concentration are solved, and then the pressure is solved. The unconditional stability and error estimations of the first order semi-discrete scheme are derived, at the same time, the unconditional stability of the first order fully discrete scheme is obtained. Some numerical experiments verify the theoretical prediction and show the effectiveness of the proposed methods.

Keywords

Acknowledgement

Research Fund from the Key Laboratory of Xinjiang Province (No.2022D04014); National Natural Science Foundation of China (No.12061075); Xinjiang Key Laboratory of Applied Mathematics (No.XJDX1401).

References

  1. M. T. Balhoff, A. Mikeli'c, and M. F. Wheeler, Polynomial filtration laws for low Reynolds number flows through porous media, Transp. Porous Media 81 (2010), no. 1, 35-60. https://doi.org/10.1007/s11242-009-9388-z 
  2. A. B. C, ibik, M. Demir, and S. Kaya, A family of second order time stepping methods for the Darcy-Brinkman equations, J. Math. Anal. Appl. 472 (2019), no. 1, 148-175. https://doi.org/10.1016/j.jmaa.2018.11.015 
  3. E. Ghasemi, S. Soleimani, and H. Bararnia, Natural convection between a circular enclosure and an elliptic cylinder using control volume based finite element method, Int. Commun. Heat Mass Transf. 39 (2012), no. 8, 1035-1044. 
  4. B. Goyeau and J. Songbe, Numerical study of double-diffusive natural convection in a porous cavity using the Darcy-Brinkman formulation, Int. J. Heat Mass Transf. 39 (1996), no. 7, 1363-1378. 
  5. B. Goyeau, J. Songbe, and D. Gobin, Numerical study of double-diffusive natural convection in a porous cavity using the Darcy-Brinkman formulation, Int. J. Heat Mass Transf. 39 (1996), no. 3, 1363-1378. 
  6. F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251-265. https://doi.org/10.1515/jnum-2012-0013 
  7. B. S. Kim, D. S. Lee, M. Y. Ha, and H. S. Yoon, A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations, Int. J. Heat Mass Transf. 51 (2007), no. 7, 1888-1906. 
  8. C. Liao and P. Huang, The modified characteristics finite element method for time-dependent Darcy-Brinkman problem, Eng. Comput. 36 (2019), no. 1, 356-376. 
  9. Y. Liu, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Math. Comput. Modelling 49 (2009), no. 7-8, 1401-1415. https://doi.org/10.1016/j.mcm.2008.11.010 
  10. M. S. Malashetty and B. S. Biradar, The onset of double diffusive reaction-convection in an anisotropic porous layer, Phys. Fluids 23 (2011), no.6, 64-102. 
  11. R. H. Nochetto and J.-H. Pyo, The gauge-Uzawa finite element method. I. The Navier-Stokes equations, SIAM J. Numer. Anal. 43 (2005), no. 3, 1043-1068. https://doi.org/10.1137/040609756 
  12. R. H. Nochetto and J.-H. Pyo, The gauge-Uzawa finite element method. II. The Boussinesq equations, Math. Models Methods Appl. Sci. 16 (2006), no. 10, 1599-1626. https://doi.org/10.1142/S0218202506001649 
  13. D. Pritchard and C. N. Richardson, The effect of temperature-dependent solubility on the onset of thermosolutal convection in a horizontal porous layer, J. Fluid Mech. 571 (2007), 59-95. https://doi.org/10.1017/S0022112006003211 
  14. J.-H. Pyo, Optimal error estimate for semi-discrete gauge-Uzawa method for the Navier-Stokes equations, Bull. Korean Math. Soc. 46 (2009), no. 4, 627-644. https://doi.org/10.4134/BKMS.2009.46.4.627 
  15. J.-H. Pyo, Error estimates for the second order semi-discrete stabilized gauge-Uzawa method for the Navier-Stokes equations, Int. J. Numer. Anal. Model. 10 (2013), no. 1, 24-41. 
  16. J.-H. Pyo and J. Shen, Gauge-Uzawa methods for incompressible flows with variable density, J. Comput. Phys. 221 (2007), no. 1, 181-197. https://doi.org/10.1016/j.jcp.2006.06.013 
  17. Y. Ren and D. Liu, Pressure correction projection finite element method for the 2D/3D time-dependent thermomicropolar fluid problem, Comput. Math. Appl. 136 (2023), 136-150. https://doi.org/10.1016/j.camwa.2023.02.011 
  18. Q. Shao, M. Fahs, and A. Younes, A high-accurate solution for Darcy-Brinkman double-diffusive convection in saturated porous media, Numer. Heat Transf. B: Fundam. 69 (2016), no. 1, 26-47. 
  19. J. Shen, On error estimates of projection methods for Navier-Stokes equations: first-order schemes, SIAM J. Numer. Anal. 29 (1992), no. 1, 57-77. https://doi.org/10.1137/0729004 
  20. R. M. Temam, Navier-Stokes equations, third edition, Studies in Mathematics and its Applications, 2, North-Holland, Amsterdam, 1984. 
  21. F. Tone, X. M. Wang, and D. Wirosoetisno, Long-time dynamics of 2d double-diffusive convection: analysis and/of numerics, Numer. Math. 130 (2015), no. 3, 541-566. https://doi.org/10.1007/s00211-014-0670-9 
  22. S. Wang and W. Tan, The onset of Darcy-Brinkman thermosolutal convection in a horizontal porous media, Phys. Lett. A 373 (2009), no. 7, 776-780. 
  23. J. Wu, J. Shen, and X. Feng, Unconditionally stable gauge-Uzawa finite element schemes for incompressible natural convection problems with variable density, J. Comput. Phys. 348 (2017), 776-789. https://doi.org/10.1016/j.jcp.2017.07.045 
  24. Y.-B. Yang and Y. L. Jiang, An explicitly uncoupled VMS stabilization finite element method for the time-dependent Darcy-Brinkman equations in double-diffusive convection, Numer. Algorithms 78 (2018), no. 2, 569-597. https://doi.org/10.1007/s11075-017-0389-7 
  25. Y. Zeng, P. Z. Huang, and Y. He, Deferred defect-correction finite element method for the Darcy-Brinkman model, ZAMM Z. Angew. Math. Mech. 101 (2021), no. 11, Paper No. e202000285, 37 pp. https://doi.org/10.1002/zamm.202000285 
  26. W. Zhu, P. Z. Huang, and K. Wang, Newton iterative method based on finite element discretization for the stationary Darcy-Brinkman equations, Comput. Math. Appl. 80 (2020), no. 12, 3098-3122. https://doi.org/10.1016/j.camwa.2020.10.020