Advertisement

Journal of Scientific Computing

, Volume 77, Issue 3, pp 1310–1338 | Cite as

An Advection-Robust Hybrid High-Order Method for the Oseen Problem

  • Joubine Aghili
  • Daniele A. Di Pietro
Article
  • 58 Downloads

Abstract

In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer \(k\geqslant 0\), the discrete velocity unknowns are vector-valued polynomials of total degree \(\leqslant \, k\) on mesh elements and faces, while the pressure unknowns are discontinuous polynomials of total degree \(\leqslant \,k\) on the mesh. From the discrete unknowns, three relevant quantities are reconstructed inside each element: a velocity of total degree \(\leqslant \,(k+1)\), a discrete advective derivative, and a discrete divergence. These reconstructions are used to formulate the discretizations of the viscous, advective, and velocity–pressure coupling terms, respectively. Well-posedness is ensured through appropriate high-order stabilization terms. We prove energy error estimates that are advection-robust for the velocity, and show that each mesh element T of diameter \(h_T\) contributes to the discretization error with an \(\mathcal {O}(h_{T}^{k+1})\)-term in the diffusion-dominated regime, an \(\mathcal {O}(h_{T}^{k+\frac{1}{2}})\)-term in the advection-dominated regime, and scales with intermediate powers of \(h_T\) in between. Numerical results complete the exposition.

Keywords

Hybrid high-order methods Oseen equations Incompressible flows Polyhedral meshes Advection-robust error estimates 

Mathematics Subject Classification

65N08 65N30 65N12 76D07 

References

  1. 1.
    Aghili, J., Boyaval, S., Di Pietro, D.A.: Hybridization of mixed high-order methods on general meshes and application to the Stokes equations. Comput. Methods Appl. Math. 15(2), 111–134 (2015).  https://doi.org/10.1515/cmam-2015-0004 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ayuso de Dios, B., Lipnikov, K., Manzini, G.: The nonconforming virtual element method. ESAIM: Math. Model. Numer. Anal. 50(3), 879–904 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Badia, S., Codina, R., Gudi, T., Guzmán, J.: Error analysis of discontinuous Galerkin methods for the Stokes problem under minimal regularity. IMA J. Numer. Anal. 34(2), 800–819 (2014).  https://doi.org/10.1093/imanum/drt022 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bassi, F., Botti, L., Colombo, A., Di Pietro, D.A., Tesini, P.: On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. 231(1), 45–65 (2012).  https://doi.org/10.1016/j.jcp.2011.08.018 MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bassi, F., Crivellini, A., Di Pietro, D.A., Rebay, S.: An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 218(2), 794–815 (2006).  https://doi.org/10.1016/j.jcp.2006.03.006 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bassi, F., Crivellini, A., Di Pietro, D.A., Rebay, S.: An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows. Comp. Fluids 36(10), 1529–1546 (2007).  https://doi.org/10.1016/j.compfluid.2007.03.012 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131(2), 267–279 (1997).  https://doi.org/10.1006/jcph.1996.5572 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Becker, R., Capatina, D., Joie, J.: Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations. Numer. Methods Partial Differ. Equ. 28(3), 1013–1041 (2012).  https://doi.org/10.1002/num.20671 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Beirao 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. 199(23), 199–214 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Beirao da Veiga, L., Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM: Math. Model. Numer. Anal. (M2AN) 51(2), 509–535 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Berlin (2013)CrossRefGoogle Scholar
  12. 12.
    Boffi, D., Di Pietro, D.A.: Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes. ESAIM: Math. Model. Numer. Anal. 52(1), 1–28 (2018).  https://doi.org/10.1051/m2an/2017036 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Botti, M., Di Pietro, D.A., Sochala, P.: A hybrid high-order method for nonlinear elasticity. SIAM J. Numer. Anal. 55(6), 2687–2717 (2017).  https://doi.org/10.1137/16M1105943 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Brezzi, F., Falk, R.S., Marini, L.D.: Basic principles of mixed virtual element methods. ESAIM: Math. Model. Numer. Anal. 48(4), 1227–1240 (2014).  https://doi.org/10.1051/m2an/2013138 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Brezzi, F., Marini, L.D., Süli, E.: Discontinuous Galerkin methods for first-order hyperbolic problems. Math. Models Methods Appl. Sci. 14(12), 1893–1903 (2004).  https://doi.org/10.1142/S0218202504003866 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Burman, E., Stamm, B.: Bubble stabilized discontinuous Galerkin method for Stokes’ problem. Math. Models Methods Appl. Sci. 20(2), 297–313 (2010).  https://doi.org/10.1142/S0218202510004234 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38, 1676–1706 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Çeşmelioğlu, A., Cockburn, B., Nguyen, N.C., Peraire, J.: Analysis of HDG methods for Oseen equations. J. Sci. Comput. 55(2), 392–431 (2013).  https://doi.org/10.1007/s10915-012-9639-y MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Çeşmelioğlu, A., Cockburn, B., Qiu, W.: Analysis of an HDG method for the incompressible Navier–Stokes equations. Math. Comput. 86, 1643–1670 (2017).  https://doi.org/10.1090/mcom/3195 CrossRefzbMATHGoogle Scholar
  20. 20.
    Cockburn, B., Di Pietro, D.A., Ern, A.: Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM: Math. Model. Numer. Anal. 50(3), 635–650 (2016).  https://doi.org/10.1051/m2an/2015051 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cockburn, B., Fu, G.: Superconvergence by \(M\)-decompositions. Part II: construction of two-dimensional finite elements. ESAIM: Math. Model. Numer. Anal. 51, 165–186 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Cockburn, B., Fu, G.: Superconvergence by \(M\)-decompositions. Part III: construction of three-dimensional finite elements. ESAIM: Math. Model. Numer. Anal. 51, 365–398 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    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).  https://doi.org/10.1137/070706616 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Cockburn, B., Gopalakrishnan, J., Sayas, F.J.: A projection-based error analysis of HDG methods. Math. Comput. 79, 1351–1367 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Cockburn, B., Hou, S., Shu, C.W.: The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545–581 (1990).  https://doi.org/10.2307/2008501 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier–Stokes equations. Math. Comput. 74(251), 1067–1095 (2005).  https://doi.org/10.1090/S0025-5718-04-01718-1 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    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).  https://doi.org/10.1007/s10915-006-9107-7 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Cockburn, B., Kanschat, G., Schötzau, D., Schwab, C.: Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40(1), 319–343 (2002).  https://doi.org/10.1137/S0036142900380121 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Cockburn, B., Lin, S.Y., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989).  https://doi.org/10.1016/0021-9991(89)90183-6 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Cockburn, B., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comput. 52(186), 411–435 (1989).  https://doi.org/10.2307/2008474 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Cockburn, B., Shu, C.W.: The Runge–Kutta local projection \(P^1\)-discontinuous-Galerkin finite element method for scalar conservation laws. RAIRO Modél. Math. Anal. Numér. 25(3), 337–361 (1991).  https://doi.org/10.1051/m2an/1991250303371 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Cockburn, B., Shu, C.W.: The Runge–Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998).  https://doi.org/10.1006/jcph.1998.5892 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Crivellini, A., D’Alessandro, V., Bassi, F.: Assessment of a high-order discontinuous Galerkin method for incompressible three-dimensional Navier–Stokes equations: benchmark results for the flow past a sphere up to \({{\rm Re}}=500\). Comput. Fluids 86, 442–458 (2013).  https://doi.org/10.1016/j.compfluid.2013.07.027 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Decuypere, R., Dibelius, G. (eds.): A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows (1997)Google Scholar
  35. 35.
    Di Pietro, D.A.: Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux. Int. J. Numer. Methods Fluids 55(8), 793–813 (2007).  https://doi.org/10.1002/fld.1495 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Di Pietro, D.A., Droniou, J.: A hybrid high-order method for Leray–Lions elliptic equations on general meshes. Math. Comput. 86(307), 2159–2191 (2017).  https://doi.org/10.1090/mcom/3180 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Di Pietro, D.A., Droniou, J.: \(W^{s, p}\)-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a hybrid high-order discretisation of Leray–Lions problems. Math. Models Methods Appl. Sci. 27(5), 879–908 (2017).  https://doi.org/10.1142/S0218202517500191 MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Di Pietro, D.A., Droniou, J., Ern, A.: A discontinuous-skeletal method for advection–diffusion–reaction on general meshes. SIAM J. Numer. Anal. 53(5), 2135–2157 (2015).  https://doi.org/10.1137/140993971 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Di Pietro, D.A., Droniou, J., Manzini, G.: Discontinuous skeletal gradient discretisation methods on polytopal meshes. J. Comput. Phys. 355, 397–425 (2018).  https://doi.org/10.1016/j.jcp.2017.11.018 MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Di Pietro, D.A., Ern, A.: Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations. Math. Comput. 79, 1303–1330 (2010).  https://doi.org/10.1090/S0025-5718-10-02333-1 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques and Applications, vol. 69. Springer, Berlin (2012)zbMATHGoogle Scholar
  42. 42.
    Di Pietro, D.A., Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015).  https://doi.org/10.1016/j.cma.2014.09.009 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Di Pietro, D.A., Ern, A., Guermond, J.L.: Discontinuous Galerkin methods for anisotropic semi-definite diffusion with advection. SIAM J. Numer. Anal. 46(2), 805–831 (2008).  https://doi.org/10.1137/060676106 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Di Pietro, D.A., Ern, A., Lemaire, S.: An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14(4), 461–472 (2014).  https://doi.org/10.1515/cmam-2014-0018 MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Di Pietro, D.A., Ern, A., Linke, A., Schieweck, F.: A discontinuous skeletal method for the viscosity-dependent Stokes problem. Comput. Methods Appl. Mech. Eng. 306, 175–195 (2016).  https://doi.org/10.1016/j.cma.2016.03.033 MathSciNetCrossRefGoogle Scholar
  46. 46.
    Di Pietro, D.A., Krell, S.: A hybrid high-order method for the steady incompressible Navier–Stokes problem. J. Sci. Comput. 74(3), 1677–1705 (2018).  https://doi.org/10.1007/s10915-017-0512-x MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Di Pietro, D.A., Lemaire, S.: An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow. Math. Comput. 84(291), 1–31 (2015).  https://doi.org/10.1090/S0025-5718-2014-02861-5 MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Di Pietro, D.A., Tittarelli, R.: Lectures from the fall 2016 thematic quarter at Institut Henri Poincaré, chap. An Introduction to Hybrid High-Order methods. SEMA-SIMAI. Springer (2017). arXiv:1703.05136 (to appear)
  49. 49.
    Droniou, J., Eymard, R., Gallouët, T., Guichard, C., Herbin, R.: The gradient discretisation method: a framework for the discretisation and numerical analysis of linear and nonlinear elliptic and parabolic problems. Maths and Applications. Springer (2017). https://hal.archives-ouvertes.fr/hal-01382358 (to appear)
  50. 50.
    Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer, New York (2004)CrossRefGoogle Scholar
  51. 51.
    Giorgiani, G., Fernández-Méndez, S., Huerta, A.: Hybridizable discontinuous Galerkin with degree adaptivity for the incompressible Navier–Stokes equations. Comput. Fluids 98, 196–208 (2014)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Girault, V., Rivière, B., Wheeler, M.F.: A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier–Stokes problems. Math. Comput. 74(249), 53–84 (2005).  https://doi.org/10.1090/S0025-5718-04-01652-7 MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Guennebaud, G., Jacob, B., et al.: Eigen v3. http://eigen.tuxfamily.org (2010)
  54. 54.
    Hansbo, P., Larson, M.G.: Piecewise divergence-free discontinuous Galerkin methods for Stokes flow. Commun. Numer. Methods Eng. 24(5), 355–366 (2008).  https://doi.org/10.1002/cnm.975 MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Herbin, R., Hubert, F.: Benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Eymard, R., Hérard, J.M. (eds.) Finite Volumes for Complex Applications, vol. V, pp. 659–692. Wiley, New York (2008)zbMATHGoogle Scholar
  56. 56.
    Karakashian, O., Katsaounis, T.: A discontinuous Galerkin method for the incompressible Navier–Stokes equations. In: Discontinuous Galerkin Methods (Newport, RI, 1999). Lecture Notes in Computational Science and Engineering, vol. 11, pp. 157–166. Springer, Berlin (2000). https://doi.org/10.1007/978-3-642-59721-3_11 Google Scholar
  57. 57.
    Kovasznay, L.I.G.: Laminar flow behind a two-dimensional grid. Math. Proc. Camb. Philos. Soc. 44(1), 58–62 (1948).  https://doi.org/10.1017/S0305004100023999 MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Mozolevski, I., Süli, E., Bösing, P.R.: Discontinuous Galerkin finite element approximation of the two-dimensional Navier–Stokes equations in stream-function formulation. Commun. Numer. Methods Eng. 23(6), 447–459 (2007).  https://doi.org/10.1002/cnm.944 MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Nguyen, N., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations. J. Comput. Phys. 230, 1147–1170 (2011)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Qiu, W., Shi, K.: A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes. IMA J. Numer. Anal. 36(4), 1943–1967 (2016)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Rivière, B., Sardar, S.: Penalty-free discontinuous Galerkin methods for incompressible Navier–Stokes equations. Math. Models Methods Appl. Sci. 24(6), 1217–1236 (2014).  https://doi.org/10.1142/S0218202513500826 MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Tavelli, M., Dumbser, M.: A staggered semi-implicit discontinuous Galerkin method for the two dimensional incompressible Navier–Stokes equations. Appl. Math. Comput. 248, 70–92 (2014).  https://doi.org/10.1016/j.amc.2014.09.089 MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Toselli, A.: \(hp\) discontinuous Galerkin approximations for the Stokes problem. Math. Models Methods Appl. Sci. 12(11), 1565–1597 (2002).  https://doi.org/10.1142/S0218202502002240 MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Ueckermann, M.P., Lermusiaux, P.F.J.: Hybridizable discontinuous Galerkin projection methods for Navier–Stokes and Boussinesq equations. J. Comput. Phys. 306, 390–421 (2016).  https://doi.org/10.1016/j.jcp.2015.11.028 MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Wihler, T.P., Wirz, M.: Mixed \(hp\)-discontinuous Galerkin FEM for linear elasticity and Stokes flow in three dimensions. Math. Models Methods Appl. Sci. 22(8), 1250016 (2012).  https://doi.org/10.1142/S0218202512500169.31 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.INRIA, Laboratoire Jean Dieudonné, CNRSUniversité Côte d’AzurNiceFrance
  2. 2.Institut Montpelliérain Alexander Grothendieck, CNRSUniv. MontpellierMontpellierFrance

Personalised recommendations