대한수학회지 (Journal of the Korean Mathematical Society)

  • 제50권5호
  • /
  • Pages.1129-1163
  • /
  • 2013
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

ON THE LINEARIZATION OF DEFECT-CORRECTION METHOD FOR THE STEADY NAVIER-STOKES EQUATIONS

  • Shang, Yueqiang (School of Mathematics and Statistics Southwest University) ;
  • Kim, Do Wan (Department of Mathematics Inha University) ;
  • Jo, Tae-Chang (Department of Mathematics Inha University)
  • 투고 : 2013.01.11
  • 발행 : 2013.09.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Based on finite element discretization, two linearization approaches to the defect-correction method for the steady incompressible Navier-Stokes equations are discussed and investigated. By applying $m$ times of Newton and Picard iterations to solve an artificial viscosity stabilized nonlinear Navier-Stokes problem, respectively, and then correcting the solution by solving a linear problem, two linearized defect-correction algorithms are proposed and analyzed. Error estimates with respect to the mesh size $h$, the kinematic viscosity ${\nu}$, the stability factor ${\alpha}$ and the number of nonlinear iterations $m$ for the discrete solution are derived for the linearized one-step defect-correction algorithms. Efficient stopping criteria for the nonlinear iterations are derived. The influence of the linearizations on the accuracy of the approximate solutions are also investigated. Finally, numerical experiments on a problem with known analytical solution, the lid-driven cavity flow, and the flow over a backward-facing step are performed to verify the theoretical results and demonstrate the effectiveness of the proposed defect-correction algorithms.

키워드

참고문헌

  1. R. Adams, Sobolev Spaces, Academic Press Inc., New York, 1975.
  2. D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations, Calcolo 21 (1984), no. 4, 337-344. https://doi.org/10.1007/BF02576171
  3. H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, Oxford University Press, Oxford, 2005.
  4. E. Erturk, T. Corke, and C. Gokcol, Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, Int. J. Numer. Methods Fluids 48 (2005), 747-774. https://doi.org/10.1002/fld.953
  5. V. J. Ervin, W. Layton, and J. M. Maubach, Adaptive defect-correction methods for viscous incompressible flow problems, SIAM J. Numer. Anal. 37 (2000), no. 4, 1165- 1185. https://doi.org/10.1137/S0036142997318164
  6. D. K. Gartling, A test problem for outflow boundary conditions-flow over a backward- facing step, Int. J. Numer. Methods Fluids 11 (1990), 953-967. https://doi.org/10.1002/fld.1650110704
  7. U. Ghia, K. Ghia, and C. Shin, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982), 387-411. https://doi.org/10.1016/0021-9991(82)90058-4
  8. V. Girault and P. A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin, 1979.
  9. V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 1986.
  10. R. Glowinski, Finite Element Methods for Incompressible Viscous Flow, in: P. G. Ciarlet and J. L. Lions (Ed.), Handbook of Numerical Analysis, Vol. IX, Numerical Methods for Fluids (Part 3), Elsevier Science Publisher, Amsterdam, 2003.
  11. Y. N. He, A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem, IMA J. Numer. Anal. 23 (2003), no. 4, 665-691. https://doi.org/10.1093/imanum/23.4.665
  12. Y. N. He and J. Li, Convergence of three iterative methods based on finite element discretization for the stationary Navier-Stokes equations, Comput. Meth. Appl. Mech. Engrg. 198 (2009), 1351-1359. https://doi.org/10.1016/j.cma.2008.12.001
  13. Y. N. He and A. W. Wang, A simplified two-level method for the steady Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 197 (2008), no. 17-18, 1568-1576. https://doi.org/10.1016/j.cma.2007.11.032
  14. F. Hecht, O. Pironneau, and K. Ohtsuka, FreeFem++, Version 3.9-2, http://www.freefem.org. [2010-12-08]
  15. J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem I: Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982), no. 2, 275-311. https://doi.org/10.1137/0719018
  16. S. Kaya, W. Layton, and B. Riviere, Subgrid stabilized defect correction methods for the Navier-Stokes equations, SIAM J. Numer. Anal. 44 (2006), no. 4, 1639-1654. https://doi.org/10.1137/050623942
  17. A. Labovschii, A defect correction method for the time-dependent Navier-Stokes equations, Numer. Methods Partial Differential Equations 25 (2009), no. 1, 1-25. https://doi.org/10.1002/num.20329
  18. W. Layton, Solution algorithm for incompressible viscous flows at high Reynolds number, Vestnik Mosk. Gos. Univ. Ser. 15 (1996), 25-35.
  19. W. Layton, H. Lee, and J. Peterson, A defect-correction method for the incompressible Navier-Stokes equations, Appl. Math. Comput. 129 (2002), no. 1, 1-19. https://doi.org/10.1016/S0096-3003(01)00026-1
  20. H. K. Lee, Analysis of a defect correction method for viscoelastic fluid flow, Comput. Math. Appl. 48 (2004), no. 7-8, 1213-1229. https://doi.org/10.1016/j.camwa.2004.10.016
  21. Q. F. Liu and Y. R. Hou, A two-level defect-correction method for Navier-Stokes equations, Bull. Aust. Math. Soc. 81 (2010), no. 3, 442-454. https://doi.org/10.1017/S0004972709000859
  22. Z. Y. Si, Y. N. He, and K. Wang, A defect-correction method for unsteady conduction convection problems I: spatial discretization, Sci. China Math. 54 (2011), no. 1, 185-204. https://doi.org/10.1007/s11425-010-4022-7
  23. Z. Y. Si, Y. N. He, and T. Zhang, A defect-correction method for unsteady conduction convection problems II: Time discretization, J. Comput. Appl. Math. 236 (2012), no. 9, 2553-2573. https://doi.org/10.1016/j.cam.2011.12.014
  24. C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids 1 (1973), no. 1, 73-100. https://doi.org/10.1016/0045-7930(73)90027-3
  25. R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
  26. K. Wang, A new defect correction method for the Navier-Stokes equations at high Reynolds numbers, Appl. Math. Comput. 216 (2010), no. 11, 3252-3264. https://doi.org/10.1016/j.amc.2010.04.050