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Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

  • Philipp W. Schroeder
  • Christoph Lehrenfeld
  • Alexander Linke
  • Gert Lube
Article

Abstract

Inf-sup stable FEM applied to time-dependent incompressible Navier–Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure–robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption \(\nabla u \in L^1(0,T;L^\infty (\varOmega ))\) which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semi-robust estimates for pointwise divergence-free \(H^1\)-conforming FEM (like Scott–Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.

Keywords

Time-dependent incompressible flow Re-semi-robust error estimates Pressure–robustness Inf-sup stable methods Exactly divergence-free FEM 

Mathematics Subject Classification

35Q30 65M15 65M60 76D17 76M10 

References

  1. 1.
    Ahmed, N., Linke, A., Merdon, C.: Towards pressure–robust mixed methods for the incompressible Navier–Stokes equations. Comput. Methods Appl. Math. (2017).  https://doi.org/10.1515/cmam-2017-0047
  2. 2.
    Arndt, D., Braack, M., Lube, G.: Finite elements for the Navier–Stokes problem with outflow condition. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds.) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol. 112, pp. 95–103. Springer, Cham (2016)CrossRefGoogle Scholar
  3. 3.
    Arndt, D., Dallmann, H., Lube, G.: Local projection FEM stabilization for the time-dependent incompressible Navier–Stokes problem. Numer. Methods Part. Differ. Equ. 31(4), 1224–1250 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ascher, U., Ruuth, S., Wetton, B.: Implicit–explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bardos, C.W., Titi, E.S.: Mathematics and turbulence: where do we stand? J. Turbul. 14(3), 42–76 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Berselli, L.C., Iliescu, T., Layton, W.J.: Mathematics of Large Eddy Simulation of Turbulent Flows. Springer, Berlin (2006)zbMATHGoogle Scholar
  7. 7.
    Bertoglio, C., Caiazzo, A., Bazilevs, Y., Braack, M., Esmaily, M., Gravemeier, V., Marsden, A., Pironneau, O., Vignon-Clementel, I.E., Wall, W.A.: Benchmark problems for numerical treatment of backflow at open boundaries. Int. J. Numer. Meth. Biomed. Eng. 34(2), e2918 (2018)CrossRefGoogle Scholar
  8. 8.
    Bertozzi, A.L.: Heteroclinic orbits and chaotic dynamics in planar fluid flows. SIAM J. Math. Anal. 19(6), 1271–1294 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Buffa, A., de Falco, C., Sangalli, G.: IsoGeometric analysis: stable elements for the 2D Stokes equation. Int. J. Numer. Methods Fluids 65(11–12), 1407–1422 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Burman, E.: Robust error estimates for stabilized finite element approximations of the two dimensional Navier–Stokes equations at high Reynolds number. Comput. Methods Appl. Mech. Eng. 288, 2–23 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Burman, E., Fernández, M.A.: Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence. Numer. Math. 107(1), 39–77 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Case, M.A., Ervin, V.J., Linke, A., Rebholz, L.G.: A connection between Scott–Vogelius and grad-div stabilized Taylor–Hood FE approximations of the Navier–Stokes equations. SIAM J. Numer. Anal. 49(4), 1461–1481 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chacón Rebollo, T., Lewandowski, R.: Mathematical and Numerical Foundations of Turbulence Models and Applications. Birkhäuser Basel, New York (2014)CrossRefzbMATHGoogle Scholar
  16. 16.
    Chrysafinos, K., Hou, L.S.: Analysis and approximations of the evolutionary Stokes equations with inhomogeneous boundary and divergence data using a parabolic saddle point formulation. ESAIM: M2AN 51(4), 1501–1526 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cockburn, B.: Two new techniques for generating exactly incompressible approximate velocities. Comput Fluid Dyn 2006, 1–11 (2009)MathSciNetGoogle Scholar
  18. 18.
    Cockburn, B.: Static condensation, hybridization, and the devising of the HDG methods. In: Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, pp. 129–177. Springer, New York (2016)Google Scholar
  19. 19.
    Cockburn, B., Gopalakrishnan, J.: Incompressible finite elements via hybridization. Part I: the Stokes system in two space dimensions. SIAM J. Numer. Anal. 43(4), 1627–1650 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cockburn, B., Gopalakrishnan, J.: Incompressible finite elements via hybridization. Part II: the Stokes system in three space dimensions. SIAM J. Numer. Anal. 43(4), 1651–1672 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cockburn, B., Gopalakrishnan, J.: The derivation of hybridizable discontinuous Galerkin methods for Stokes flow. SIAM J. Numer. Anal. 47(2), 1092–1125 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cockburn, B., Kanschat, G., Schötzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes equations. J. Sci. Comput. 31(1–2), 61–73 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Cousins, B.R., Le Borne, S., Linke, A., Rebholz, L.G., Wang, Z.: Efficient linear solvers for incompressible flow simulations using Scott–Vogelius finite elements. Numer. Methods Part. Differ. Equ. 29(4), 1217–1237 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Dallmann, H., Arndt, D.: Stabilized finite element methods for the Oberbeck–Boussinesq model. J. Sci. Comput. 69(1), 244–273 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  27. 27.
    Durst, F.: Fluid Mechanics: An Introduction to the Theory of Fluid Flows. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  28. 28.
    Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
  29. 29.
    Evans, J.A.: Divergence-free B-spline discretizations for viscous incompressible flows. Ph.D. thesis, The University of Texas at Austin (2011)Google Scholar
  30. 30.
    Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations. Math. Model Methods Appl. Sci. 23(8), 1421–1478 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations. J. Comput. Phys. 241, 141–167 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    de Frutos, J., García-Archilla, B., John, V., Novo, J.: Semi-robust local projection stabilization for non inf-sup stable discretizations of the evolutionary Navier–Stokes equations. arXiv:1709.01011 [math.NA] (2017)
  33. 33.
    de Frutos, J., García-Archilla, B., John, V., Novo, J.: Analysis of the grad-div stabilization for the time-dependent Navier–Stokes equations with inf-sup stable finite elements. Adv. Comput. Math. 44(1), 195–225 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Girault, V., Nochetto, R.H., Scott, L.R.: Max-norm estimates for Stokes and Navier–Stokes approximations in convex polyhedra. Numer. Math. 131(4), 771–822 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Guzmán, J., Shu, C.W., Sequeira, F.A.: H(div) conforming and DG methods for incompressible Euler’s equations. IMA J. Numer. Anal. 37(4), 1733–1771 (2017)MathSciNetGoogle Scholar
  36. 36.
    Henshaw, W.D., Kreiss, H.O., Reyna, L.G.: Smallest scale estimates for the Navier–Stokes equations for incompressible fluids. Arch. Ration. Mech. Anal. 112(1), 21–44 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2007)zbMATHGoogle Scholar
  38. 38.
    Jenkins, E.W., John, V., Linke, A., Rebholz, L.G.: On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40(2), 491–516 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    John, V.: Finite Element Methods for Incompressible Flow Problems. Springer, New York (2016)CrossRefzbMATHGoogle Scholar
  40. 40.
    John, V., Knobloch, P., Novo, J.: Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story? Comput. Vis. Sci. (2018).  https://doi.org/10.1007/s00791-018-0290-5
  41. 41.
    John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492–544 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Lederer, P.L., Lehrenfeld, C., Schöberl, J.: Hybrid discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part I. (2017). arXiv:1707.02782 [math.NA]
  43. 43.
    Lederer, P.L., Linke, A., Merdon, C., Schöberl, J.: Divergence-free reconstruction operators for pressure–robust Stokes discretizations with continuous pressure finite elements. SIAM J. Numer. Anal. 55(3), 1291–1314 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Lederer, P.L., Schöberl, J.: Polynomial robust stability analysis for \(H(\operatorname{div})\)-conforming finite elements for the Stokes equations. IMA J. Numer. Anal. (2017).  https://doi.org/10.1093/imanum/drx051
  45. 45.
    Lehrenfeld, C.: Hybrid discontinuous Galerkin methods for solving incompressible flow problems. Master’s thesis, RWTH Aachen (2010)Google Scholar
  46. 46.
    Lehrenfeld, C., Schöberl, J.: High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput. Methods Appl. Mech. Eng. 307, 339–361 (2016)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Linke, A.: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Eng. 268, 782–800 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Linke, A., Matthies, G., Tobiska, L.: Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors. ESAIM: M2AN 50(1), 289–309 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Linke, A., Merdon, C.: On velocity errors due to irrotational forces in the Navier–Stokes momentum balance. J. Comput. Phys. 313, 654–661 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Linke, A., Merdon, C.: Pressure–robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 311, 304–326 (2016)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Lube, G., Arndt, D., Dallmann, H.: Understanding the limits of inf-sup stable Galerkin-FEM for incompressible flows. In: Knobloch, P. (ed.) Boundary and Interior Layers, Computational and Asymptotic Methods—BAIL 2014. Lecture Notes in Computational Science and Engineering, vol. 108, pp. 147–169. Springer, Cham (2015)Google Scholar
  52. 52.
    Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  53. 53.
    Natale, A., Cotter, C.J.: A variational H(div) finite-element discretization approach for perfect incompressible fluids. IMA J. Numer. Anal. (2017).  https://doi.org/10.1093/imanum/drx033
  54. 54.
    Oberai, A.A., Liu, J., Sondak, D., Hughes, T.J.R.: A residual based eddy viscosity model for the large eddy simulation of turbulent flows. Comput. Methods Appl. Mech. Eng. 282, 54–70 (2014)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Oikawa, I.: A hybridized discontinuous Galerkin method with reduced stabilization. J. Sci. Comput. 65(1), 327–340 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Oikawa, I.: Analysis of a reduced-order HDG method for the Stokes equations. J. Sci. Comput. 67(2), 475–492 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Rhebergen, S., Wells, G.N.: Analysis of a hybridized/interface stabilized finite element method for the Stokes equations. SIAM J. Numer. Anal. 55(4), 1982–2003 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Rhebergen, S., Wells, G.N.: A hybridizable discontinuous Galerkin method for the Navier-Stokes equations with pointwise divergence-free velocity field. J. Sci. Comput. (2018).  https://doi.org/10.1007/s10915-018-0671-4
  59. 59.
    Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. SIAM (2008)Google Scholar
  60. 60.
    Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems, 2nd edn. Springer, Berlin (2008)zbMATHGoogle Scholar
  61. 61.
    Schlichting, H., Gersten, K.: Boundary-Layer Theory, 8th edn. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  62. 62.
    Schöberl, J.: C++11 Implementation of Finite Elements in NGSolve. ASC Report 30/2014, Institute for Analysis and Scientific Computing, Vienna University of Technology (2014). https://www.asc.tuwien.ac.at/~schoeberl/wiki/publications/ngs-cpp11.pdf
  63. 63.
    Schöberl, J., Zaglmayr, S.: High order Nédélec elements with local complete sequence properties. COMPEL 24(2), 374–384 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Schroeder, P.W., Lube, G.: Divergence-free H(div)-FEM for time-dependent incompressible flows with applications to high Reynolds number vortex dynamics. J. Sci. Comput. 75(2), 830–858 (2018)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Schroeder, P.W., Lube, G.: Pressure–robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier–Stokes flows. J. Numer. Math. 25(4), 249–276 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Tritton, D.J.: Physical Fluid Dynamics, 2nd edn. Oxford University Press, New York (1988)zbMATHGoogle Scholar
  67. 67.
    Zaglmayr, S.: High Order Finite Element Methods for Electromagnetic Field Computation. Ph.D. thesis, Johannes Kepler University Linz (2006)Google Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2018

Authors and Affiliations

  1. 1.Institute for Numerical and Applied MathematicsGeorg-August-University GöttingenGöttingenGermany
  2. 2.Weierstrass InstituteBerlinGermany

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