Journal of Scientific Computing

, Volume 65, Issue 3, pp 1189–1216 | Cite as

Simple Finite Element Numerical Simulation of Incompressible Flow Over Non-rectangular Domains and the Super-Convergence Analysis

  • Yunhua Xue
  • Cheng Wang
  • Jian-Guo Liu


In this paper, we apply a simple finite element numerical scheme, proposed in an earlier work (Liu in Math Comput 70(234):579–593, 2000), to perform a high resolution numerical simulation of incompressible flow over an irregular domain and analyze its boundary layer separation. Compared with many classical finite element fluid solvers, this numerical method avoids a Stokes solver, and only two Poisson-like equations need to be solved at each time step/stage. In addition, its combination with the fully explicit fourth order Runge–Kutta (RK4) time discretization enables us to compute high Reynolds number flow in a very efficient way. As an application of this robust numerical solver, the dynamical mechanism of the boundary layer separation for a triangular cavity flow with Reynolds numbers \(Re=10^4\) and \(Re=10^5\), including the precise values of bifurcation location and critical time, are reported in this paper. In addition, we provide a super-convergence analysis for the simple finite element numerical scheme, using linear elements over a uniform triangulation with right triangles.


Incompressible flows Boundary layer separation  Structural bifurcation Simple finite element method  Super-convergence analysis 

Mathematics Subject Classification

65N30 65M12 35Q30 76D05 



The authors greatly appreciate many helpful discussions with Wenbin Chen, Sigal Gottlieb, Yuan Liu and Chi-Wang Shu, in particular for their insightful suggestion and comments. This work is supported in part by the NSF DMS-1115420 NSF DMS-1418689 (C. Wang), NSFC 11271281 (C. Wang), NSFC 11171168 (Y. Xue) and the fund by China Scholarship Council (Y. Xue). Y. Xue thanks University of Massachusetts Dartmouth, for support during his visit. C. Wang also thanks Shanghai Center for Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, for support during his visit.


  1. 1.
    Auteri, F., Parolini, N., Quartapelle, L.: Numerical investigation on the stability of singular driven cavity flow. J. Comput. Phys. 183, 1–25 (2002)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Barragy, E., Carey, G.F.: Stream function-vorticity driven cavity solution using \(p\) finite elements. Comput. Fluids 26(5), 453–468 (1997)MATHCrossRefGoogle Scholar
  3. 3.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)MATHCrossRefGoogle Scholar
  4. 4.
    Brenner, S., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2010)Google Scholar
  5. 5.
    Bruneau, C.-H., Saad, M.: The 2D lid-driven cavity problem revisited. Comput. Fluids 35, 326–348 (2006)MATHCrossRefGoogle Scholar
  6. 6.
    Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics. Springer, Berlin (1997)Google Scholar
  7. 7.
    Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)Google Scholar
  8. 8.
    Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier–Stokes equations. Math. Comput. 74, 1067–1095 (2005)MATHCrossRefGoogle Scholar
  9. 9.
    Coupez, T., Hachem, E.: Solution of high-Reynolds incompressible flow with stabilized finite element and adaptive anisotropic meshing. Comput. Methods Appl. Mech. Eng. 267, 65–85 (2013)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Di, Y., Li, R., Tang, T., Zhang, P.: Moving mesh finite element methods for the incompressible Navier–Stokes equations. SIAM J. Sci. Comput. 26, 1036–1056 (2005)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    E, W., Liu, J.-G.: Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys. 126, 122–138 (1996)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    E, W., Liu, J.-G.: Vorticity boundary condition and related issues for finite difference schemes. J. Comput. Phys. 124, 368–382 (1996)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Erturk, E., Corke, T., Gokcol, C.: Numerical solutions of 2D steady incompressible driven cavity flow at high Reynolds numbers. Int. J. Numer. Methods Fluids 48, 747–774 (2005)MATHCrossRefGoogle Scholar
  14. 14.
    Erturk, E., Gokcol, O.: Fine grid numerical solutions of triangular cavity flow. Eur. Phys. J. Appl. Phys. 38, 97–105 (2007)CrossRefGoogle Scholar
  15. 15.
    China Fegensoft software company.
  16. 16.
    Center for Applied Scientific Computing of Lawrence Livermore National Lab.
  17. 17.
    Gargano, F., Sammartino, M., Sciacca, V.: High Reynolds number Navier–Stokes solutions and boundary layer separation induced by a rectilinear vortex. Comput. Fluids 52, 73–91 (2011)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Ge, L., Sotiropoulos, F.: A numerical method for solving the 3D unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries. J. Comput. Phys. 225, 1782–1809 (2007)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    George, A., Huang, L.C., Tang, W., Wu, Y.: Numerical simulation of unsteady incompressible flow \((Re\le 9500)\) on the curvilinear half-staggered mesh. SIAM J. Sci. Comput. 21(6), 2331–2351 (2000)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Ghia, U., Ghia, K., Shin, C.: High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J. Comput. Phys 48, 387–411 (1982)MATHCrossRefGoogle Scholar
  21. 21.
    Ghil, M., Liu, J.-G., Wang, C., Wang, S.: Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow. Phys. D 197, 149–173 (2004)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Ghil, M., Ma, T., Wang, S.: Structure of 2-D incompressible flows with the Dirichlet boundary conditions. Discrete Contin. Dyn. Syst. B 1, 29–41 (2001)CrossRefGoogle Scholar
  23. 23.
    Ghil, M., Ma, T., Wang, S.: Structural bifurcation of 2-D nondivergent flows with Dirichlet boundary condition: applications to boundary-layer separation. SIAM J. Appl. Math. 65(5), 1576–1596 (2005)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Girault, V., Raviart, P.: Finite Element Method for Navier–Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)CrossRefGoogle Scholar
  25. 25.
    Girault, V., Riviére, B., Wheeler, M.F.: A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier–Stokes problems. Math. Comput. 74, 53–84 (2005)MATHCrossRefGoogle Scholar
  26. 26.
    Goldstein, S.: Modern Developments in Fluid Fynamics. Dover Publications, New York (1965)Google Scholar
  27. 27.
    Gonzalez, L.M., Ahmed, M., Kuhnen, J.: Three-dimensional flow instability in a lid-driven isosceles triangular cavity. J. Fluid Mech. 675, 369–396 (2011)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Hachem, E., Rivaux, B., Kloczko, T., Digonnet, H., Coupez, T.: Stabilized finite element method for incompressible flows with high Reynolds number. J. Comput. Phys. 229, 8643–8665 (2010)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    He, Y.: Euler implicit/explicit iterative scheme for the stationary Navier–Stokes equations. Numer. Math. 123, 67–96 (2013)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Heywood, J., Rannacher, R.: 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, 275–310 (1982)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Holdeman, J.T.: A Hermite finite element for incompressible fluid flow. Int. J. Numer. Methods Fluids 64, 376–408 (2010)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Huang, Y., Liu, J.-G., Wang, W.: A generalized MAC scheme on curvilinear domains. SIAM J. Sci. Comput. 35(5), 953–986 (2013)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Jagannathan, A., Mohan, R., Dhanak, M.: A spectral method for the triangular cavity flow. Comput. Fluids 95, 40–48 (2014)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Johnston, H., Liu, J.-G.: A finite difference scheme for incompressible flow based on local pressure boundary conditions. J. Comput. Phys. 180, 120–154 (2002)MATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Johnston, H., Liu, J.-G.: Accurate, stable and efficient Navier–Stokes solvers based on explicit treatment of the pressure term. J. Comput. Phys. 199, 221–259 (2004)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Kalita, J.C., Sen, S.: The biharmonic approach for unsteady flow past an impulsively started circular cylinder. Commun. Comput. Phys. 12, 1163–1182 (2012)MathSciNetGoogle Scholar
  37. 37.
    Koumoutsakost, P., Leonard, A.: High-resolution simulations of the flow around an impulsively started cylinder using vortex methods. J. Fluid Mech. 296, 1–38 (1995)CrossRefGoogle Scholar
  38. 38.
    Lai, M.-J., Wenston, P.: Bivariate splines for fluid flows. Comput. Fluids 33, 1047–1073 (2004)MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Layton, W., Lee, H., Peterson, J.: A defect-correction method for the incompressible Navier–Stokes equations. Appl. Math. Comput. 129, 1–19 (2002)MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Li, M., Tang, T.: Steady viscous flow in a triangular cavity by efficient numerical techniques. Comput. Math. Appl. 31(10), 55–65 (1996)MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Li, Z., Wang, C.: A fast finite difference method for solving Navier–Stokes equations on irregular domains. Commun. Math. Sci. 1(1), 181–197 (2003)CrossRefGoogle Scholar
  42. 42.
    Lin, Q., Yan, N.: Structure and Analysis for Efficient Finite Element Methods. Publishers of Hebei University, Baoding (1996). (in Chinese)Google Scholar
  43. 43.
    Liu, J.-G., E, W.: Simple finite element methods in vorticity formulation for incompressible flows. Math. Comput. 70(234), 579–593 (2000)Google Scholar
  44. 44.
    Oleinik, O.A., Samokhin, V.N.: Mathematical Models in Boundary Layer Theory. Chapman and Hall, London (1999)MATHGoogle Scholar
  45. 45.
    Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38, 437–445 (1982)MATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Rhebergen, S., Cockburn, B., van der Vegt, J.J.W.: A space–time discontinuous Galerkin method for the incompressible Navier–Stokes equations. J. Comput. Phys. 233, 339–358 (2013)MATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Ribbens, C.J., Watson, L.T., Wang, C.-Y.: Steady viscous flow in a triangular cavity. J. Comput. Phys. 112(1), 173–181 (1994)MATHCrossRefGoogle Scholar
  48. 48.
    Shahbazi, K., Fischer, P.F., Ethier, C.R.: A high-order discontinuous Galerkin method for the unsteady incompressible Navier–Stokes equations. J. Comput. Phys. 222, 391–407 (2007)MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Wang, K., Wong, Y.S.: Error correction method for Navier–Stokes equations at high Reynolds numbers. J. Sci. Comput. 255, 245–265 (2013)MathSciNetGoogle Scholar
  50. 50.
    Yan, N.: Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Methods. Science Press, China (2005)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.School of Mathematics Sciences and LPMCNankai UniversityTianjinChina
  2. 2.Department of MathematicsUniversity of Massachusetts DartmouthNorth DartmouthUSA
  3. 3.Department of Physics and Department of MathematicsDuke UniversityDurhamUSA

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