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A Virtual Element method for non-Newtonian fluid flows

Published 6 Mar 2024 in math.NA and cs.NA | (2403.03886v1)

Abstract: In this paper, we design and analyze a Virtual Element discretization for the steady motion of non-Newtonian, incompressible fluids. A specific stabilization, tailored to mimic the monotonicity and boundedness properties of the continuous operator, is introduced and theoretically investigated. The proposed method has several appealing features, including the exact enforcement of the divergence free condition and the possibility of making use of fully general polygonal meshes. A complete well-posedness and convergence analysis of the proposed method is presented under mild assumptions on the non-linear laws, encompassing common examples such as the Carreau--Yasuda model. Numerical experiments validating the theoretical bounds as well as demonstrating the practical capabilities of the proposed formulation are presented.

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References (72)
  1. A virtual element discretization for the time dependent Navier-Stokes equations in stream-function formulation. ESAIM Math. Model. Numer. Anal., 55(5):2535–2566, 2021.
  2. The Morley-type virtual element method for the Navier-Stokes equations in stream-function form. Comput. Methods Appl. Mech. Engrg., 419:Paper No. 116573, 28, 2024.
  3. High-order discontinuous Galerkin methods on polyhedral grids for geophysical applications: seismic wave propagation and fractured reservoir simulations. In Polyhedral methods in geosciences, volume 27 of SEMA SIMAI Springer Ser., pages 159–225. Springer, Cham, [2021] ©2021.
  4. h⁢pℎ𝑝hpitalic_h italic_p-version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM J. Sci. Comput., 35(3):A1417–A1439, 2013.
  5. Virtual element method for the Navier-Stokes equation coupled with the heat equation. IMA J. Numer. Anal., 43(6):3396–3429, 2023.
  6. A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal., 52(1):386–404, 2014.
  7. The Virtual Element Method and its Applications. SEMA SIMAI Springer series 31, Springer International Publishing, 2022.
  8. J. Baranger and K. Najib. Analyse numérique des écoulements quasi-newtoniens dont la viscosité obéit à la loi puissance ou la loi de carreau. Numer. Math., 58(1):35–49, 1990.
  9. Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow. Numer. Math., 68(4):437–456, 1994.
  10. Finite element error analysis of a quasi-Newtonian flow obeying the Carreau or power law. Numer. Math., 64(4):433–453, 1993.
  11. H. Beirão da Veiga. On the global regularity of shear thinning flows in smooth domains. J. Math. Anal. Appl., 349(2):335–360, 2009.
  12. The virtual element method. Acta Numer., 32:123–202, 2023.
  13. Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes. Comput. Methods Appl. Mech. Engrg., 397:Paper No. 115061, 31, 2022.
  14. High-order virtual element method on polyhedral meshes. Comput. Math. Appl., 74(5):1110–1122, 2017.
  15. The Stokes complex for virtual elements in three dimensions. Math. Models Methods Appl. Sci., 30(3):477–512, 2020.
  16. Vorticity-stabilized virtual elements for the Oseen equation. Mathematical Models and Methods in Applied Sciences, 31(14):3009–3052, 2021.
  17. The mimetic finite difference method for elliptic problems, volume 11 of MS&A. Modeling, Simulation and Applications. Springer, Cham, 2014.
  18. Divergence free Virtual Elements for the Stokes problem on polygonal meshes. ESAIM: Math. Model. Numer. Anal. (M2AN), 51(2):509–535, 2017.
  19. Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM Math. Model. Numer. Anal., 51(2):509–535, 2017.
  20. Virtual elements for the Navier-Stokes problem on polygonal meshes. SIAM J. Numer. Anal., 56(3):1210–1242, 2018.
  21. The Stokes complex for virtual elements with application to Navier-Stokes flows. J. Sci. Comput., 81(2):990–1018, 2019.
  22. L. Beirão da Veiga and G. Vacca. Sharper error estimates for virtual elements and a bubble-enriched version. SIAM J. Numer. Anal., 60(4):1853–1878, 2022.
  23. Basic principles of virtual element methods. Math. Models Methods Appl. Sci., 23(1):199–214, 2013.
  24. Stability analysis for the virtual element method. Math. Mod.and Meth. in Appl. Sci., 27(13):2557–2594, 2017.
  25. A fully-discrete virtual element method for the nonstationary boussinesq equations in stream-function form. Computer Methods in Applied Mechanics and Engineering, 408:115947, 2023.
  26. On the finite element approximation of p𝑝pitalic_p-Stokes systems. SIAM J. Numer. Anal., 50(2):373–397, 2012.
  27. Existence of strong solutions for incompressible fluids with shear dependent viscosities. J. Math. Fluid Mech., 12(1):101–132, 2010.
  28. BDDC Preconditioners for Virtual Element Approximations of the Three-Dimensional Stokes Equations. SIAM J. Sci. Comput., 46(1):A156–A178, 2024.
  29. T. Bevilacqua and S. Scacchi. BDDC preconditioners for divergence free virtual element discretizations of the Stokes equations. J. Sci. Comput., 92(2):Paper No. 63, 27, 2022.
  30. Mixed finite element methods and applications, volume 44 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2013.
  31. M. Bogovskiĭ. Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR, 248(5):1037–1040, 1979.
  32. A Hybrid High-Order method for creeping flows of non-Newtonian fluids. ESAIM: Math. Model Numer. Anal., 55(5):2045–2073, 2021.
  33. Some estimates for virtual element methods. Comput. Methods Appl. Math., 17(4):553–574, 2017.
  34. The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics. Springer, New York, third edition, 2008.
  35. Virtual element methods on meshes with small edges or faces. Math. Models Methods Appl. Sci., 28(7):1291–1336, 2018.
  36. H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer New York, 2010.
  37. E. Cáceres and G. N. Gatica. A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem. IMA J. Numer. Anal., 37(1):296–331, 2017.
  38. A mixed virtual element method for quasi-Newtonian Stokes flows. SIAM J. Numer. Anal., 56(1):317–343, 2018.
  39. h⁢pℎ𝑝hpitalic_h italic_p-version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci., 24(10):2009–2041, 2014.
  40. The nonconforming virtual element method for the Stokes equations. SIAM J. Numer. Anal., 54(6):3411–3435, 2016.
  41. A Hybrid High-Order method for incompressible flows of non-newtonian fluids with power-like convective behaviour. IMA Journal of Numerical Analysis, 43(1):144–186, 12 2021.
  42. p𝑝pitalic_p- and h⁢pℎ𝑝hpitalic_h italic_p-virtual elements for the Stokes problem. Adv. Comput. Math., 47(2):Paper No. 24, 31, 2021.
  43. P. G. Ciarlet and P. Ciarlet. Another approach to linearized elasticity and a new proof of korn’s inequality. Mathematical Models and Methods in Applied Sciences, 15(02):259–271, 2005.
  44. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal., 47(2):1319–1365, 2009.
  45. A projection-based error analysis of HDG methods. Mathematics of Computation, 79(271):1351–1367, 2010.
  46. K. Deimling. Nonlinear functional analysis. Springer-Verlag, Berlin, 1985.
  47. D. A. Di Pietro and J. Droniou. The hybrid high-order method for polytopal meshes, volume 19 of MS&A. Modeling, Simulation and Applications. Springer, Cham, [2020] ©2020. Design, analysis, and applications.
  48. D. A. Di Pietro and A. Ern. A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Meth. Appl. Mech. Engrg., 283:1–21, 2015.
  49. 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.
  50. L. Diening and F. Ettwein. Fractional estimates for non-differentiable elliptic systems with general growth. Forum Math., 20(3):523–556, 2008.
  51. Q. Du and M.D. Gunzuburger. Finite-element approximation of a ladyzhenskaya model for stationary incompressible viscous flows. SIAM J. Numer. Analysis, 27:1–19, 1990.
  52. D. Frerichs and C. Merdon. Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem. IMA J. Numer. Anal., 42(1):597–619, 2022.
  53. A mixed virtual element method for the Navier-Stokes equations. Math. Models Methods Appl. Sci., 28(14):2719–2762, 2018.
  54. An LpsuperscriptL𝑝{\rm L}^{p}roman_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT spaces-based mixed virtual element method for the two-dimensional Navier-Stokes equations. Math. Models Methods Appl. Sci., 31(14):2937–2977, 2021.
  55. Analysis of an augmented HDG method for a class of quasi-newtonian Stokes flows. J. Sci. Comput., 65(3):1270–1308, 2015.
  56. G. Geymonat and P. M. Suquet. Functional spaces for Norton-Hoff materials. Math. Methods Appl. Sci., 8(2):206–222, 1986.
  57. A. Hirn. Approximation of the p𝑝pitalic_p-Stokes equations with equal-order finite elements. J. Math. Fluid Mech., 15(1):65–88, 2013.
  58. A. Kaltenbach and M. RůžIčka. A local discontinuous Galerkin approximation for the p-Navier-Stokes system, part i: Convergence analysis. SIAM Journal on Numerical Analysis, 61(4):1613–1640, 2023.
  59. C. Kreuzer and E. Süli. Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. ESAIM: Math. Model. Numer. Anal., 50(5):1333–1369, 2016.
  60. Quasi-optimal and pressure robust discretizations of the stokes equations by moment- and divergence-preserving operators. Computational Methods in Applied Mathematics, 21(2):423–443, 2021.
  61. F. Lepe and G. Rivera. A virtual element approximation for the pseudostress formulation of the Stokes eigenvalue problem. Comput. Methods Appl. Mech. Engrg., 379:Paper No. 113753, 21, 2021.
  62. P. Lewintan and P. Neff. Nečas–Lions lemma revisited: An Lpsuperscript𝐿𝑝{L}^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-version of the generalized Korn inequality for incompatible tensor fields. Math. Models Methods Appl. Sci., 44(14):11392–11403, 2021.
  63. A Least-Squares stabilization virtual element method for the Stokes problem on polygonal meshes. Int. J. Numer. Anal. Model., 19(5):685–708, 2022.
  64. On stabilized equal-order virtual element methods for the Navier-Stokes equations on polygonal meshes. Comput. Math. Appl., 154:267–286, 2024.
  65. Generalizations of SIP methods to systems with p𝑝pitalic_p-structure. IMA Journal of Numerical Analysis, 38(3):1420–1451, 09 2017.
  66. Stability and interpolation properties for Stokes-like virtual element spaces. J. Sci. Comput., 94(56), 2023.
  67. D. Sandri. Sur l’approximation numérique des écoulements quasi-newtoniens dont la viscosité suit la loi puissance ou la loi de carreau. M2AN Math. Model. Numer. Anal., 27(2):131–155, 1993.
  68. PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct. Multidiscip. Optim., 45(3):309–328, 2012.
  69. G. Vacca. An H1superscript𝐻1H^{1}italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-conforming virtual element for Darcy and Brinkman equations. Math. Models Methods Appl. Sci., 28(1):159–194, 2018.
  70. G. Wang and Y. Wang. Least-Squares Virtual Element Method for Stokes Problems on Polygonal Meshes. J. Sci. Comput., 98(2):Paper no. 46, 2024.
  71. L. Wang. On Korn’s inequality. Journal of Computational Mathematics, 21(3):321–324, 2003.
  72. Shear flow properties of concentrated solutions of linear and star branched polystyrenes. Rheologica Acta, 20(2):163–178, 3 1981.
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