Skip to main content
Log in

On the analysis and approximation of some models of fluids over weighted spaces on convex polyhedra

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We study the Stokes problem over convex polyhedral domains on weighted Sobolev spaces. The weight is assumed to belong to the Muckenhoupt class \(\varvec{A}_{\varvec{q}}\) for \(\varvec{q} \in (1,\varvec{\infty })\). We show that the Stokes problem is well-posed for all \(\varvec{q}\). In addition, we show that the finite element Stokes projection is stable on weighted spaces. With the aid of these tools, we provide well-posedness and approximation results to some classes of non-Newtonian fluids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Frehse, J., Málek, J., Steinhauer, M.: On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34(5), 1064–1083 (2003). https://doi.org/10.1137/S0036141002410988

    Article  MathSciNet  MATH  Google Scholar 

  2. Bulíček, M., Burczak, J., Schwarzacher, S.: A unified theory for some non-Newtonian fluids under singular forcing. SIAM J. Math. Anal. 48(6), 4241–4267 (2016)

    Article  MathSciNet  Google Scholar 

  3. Schumacher, K.: The stationary Navier–Stokes equations in weighted Bessel-potential spaces. J. Math. Soc. Japan 61(1), 1–38 (2009)

    Article  MathSciNet  Google Scholar 

  4. Bothe, D., Prüss, J.: \(L_P\)-theory for a class of non-Newtonian fluids. SIAM J. Math. Anal. 39(2), 379–421 (2007). https://doi.org/10.1137/060663635

    Article  MathSciNet  MATH  Google Scholar 

  5. Beirão da Veiga, H.: On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem. J. Eur. Math. Soc. (JEMS) 11(1), 127–167 (2009). https://doi.org/10.4171/JEMS/144

    Article  MathSciNet  MATH  Google Scholar 

  6. Ladyženskaja, O.A.: Matematicheskie Voprosy Dinamiki Vyazkoĭ Neszhimaemoĭ Zhidkosti, p. 203. Gosudarstv. Izdat. Fiz.-Mat. Lit, Moscow (1961)

    Google Scholar 

  7. Ladyženskaja, O.A.: Modifications of the Navier-Stokes equations for large gradients of the velocities. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 126–154 (1968)

  8. Ladyženskaja, O.A.: New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems. Trudy Mat. Inst. Steklov. 102, 85–104 (1967)

    MathSciNet  Google Scholar 

  9. Du, Q., Gunzburger, M.D.: Analysis of a Ladyzhenskaya model for incompressible viscous flow. J. Math. Anal. Appl. 155(1), 21–45 (1991). https://doi.org/10.1016/0022-247X(91)90024-T

    Article  MathSciNet  MATH  Google Scholar 

  10. Frehse, J., Málek, J., Steinhauer, M.: An existence result for fluids with shear dependent viscosity—steady flows. In: Proceedings of the Second World Congress of Nonlinear Analysts, Part 5 (Athens, 1996), vol. 30, pp. 3041–3049 (1997). https://doi.org/10.1016/S0362-546X(97)00392-1

  11. Málek, J., Rajagopal, K.R., Růžička, M.: Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models Methods Appl. Sci. 5(6), 789–812 (1995). https://doi.org/10.1142/S0218202595000449

    Article  MathSciNet  MATH  Google Scholar 

  12. Du, Q., Gunzburger, M.: Finite-element approximations of a Ladyzhenskaya model for stationary incompressible viscous flow. SIAM J. Numer. Anal. 27(1), 1–19 (1990). https://doi.org/10.1137/0727001

    Article  MathSciNet  MATH  Google Scholar 

  13. Baranger, J., Najib, K.: 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). https://doi.org/10.1007/BF01385609

    Article  MathSciNet  MATH  Google Scholar 

  14. Sandri, D.: Sur l’approximation numérique des écoulements quasi-newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau. RAIRO Modél. Math. Anal. Numér. 27(2), 131–155 (1993). https://doi.org/10.1051/m2an/1993270201311

    Article  MathSciNet  MATH  Google Scholar 

  15. Barrett, J.W., Liu, W.B.: Finite element error analysis of a quasi-Newtonian flow obeying the Carreau or power law. Numer. Math. 64(4), 433–453 (1993). https://doi.org/10.1007/BF01388698

    Article  MathSciNet  MATH  Google Scholar 

  16. Barrett, J.W., Liu, W.B.: Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow. Numer. Math. 68(4), 437–456 (1994). https://doi.org/10.1007/s002110050071

    Article  MathSciNet  MATH  Google Scholar 

  17. Belenki, L., Berselli, L.C., Diening, L., Růžička, M.: On the finite element approximation of \(p\)-Stokes systems. SIAM J. Numer. Anal. 50(2), 373–397 (2012). https://doi.org/10.1137/10080436X

    Article  MathSciNet  MATH  Google Scholar 

  18. Hirn, A.: Finite element approximation of singular power-law systems. Math. Comp. 82(283), 1247–1268 (2013). https://doi.org/10.1090/S0025-5718-2013-02668-3

    Article  MathSciNet  MATH  Google Scholar 

  19. Durán, R.G., Otárola, E., Salgado, A.J.: Stability of the Stokes projection on weighted spaces and applications. Math. Comp. 89(324), 1581–1603 (2020). https://doi.org/10.1090/mcom/3509

    Article  MathSciNet  MATH  Google Scholar 

  20. Rappaz, J., Rochat, J.: On non-linear Stokes problems with viscosity depending on the distance to the wall. C. R. Math. Acad. Sci. Paris 354(5), 499–502 (2016). https://doi.org/10.1016/j.crma.2016.01.022

    Article  MathSciNet  MATH  Google Scholar 

  21. Layton, W.: Energy dissipation in the Smagorinsky model of turbulence. Appl. Math. Lett. 59, 56–59 (2016). https://doi.org/10.1016/j.aml.2016.03.008

    Article  MathSciNet  MATH  Google Scholar 

  22. Duoandikoetxea, J.: Fourier Analysis. Graduate Studies in Mathematics, vol. 29, p. 222. American Mathematical Society, Providence, RI (2001). Translated and revised from the 1995 Spanish original by David Cruz-Uribe

  23. Gol’dshtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Am. Math. Soc. 361(7), 3829–3850 (2009). https://doi.org/10.1090/S0002-9947-09-04615-7

    Article  MathSciNet  MATH  Google Scholar 

  24. Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics, vol. 1736, p. 173. Springer, Berlin (2000). https://doi.org/10.1007/BFb0103908

    Book  MATH  Google Scholar 

  25. Diening, L., Růžička, M., Schumacher, K.: A decomposition technique for John domains. Ann. Acad. Sci. Fenn. Math. 35(1), 87–114 (2010). https://doi.org/10.5186/aasfm.2010.3506

    Article  MathSciNet  MATH  Google Scholar 

  26. Farwig, R., Sohr, H.: Weighted \(L^q\)-theory for the Stokes resolvent in exterior domains. J. Math. Soc. Japan 49(2), 251–288 (1997)

    Article  MathSciNet  Google Scholar 

  27. Otárola, E., Salgado, A.J.: The Poisson and Stokes problems on weighted spaces in Lipschitz domains and under singular forcing. J. Math. Anal. Appl. 471(1), 599–612 (2019). https://doi.org/10.1016/j.jmaa.2018.10.094

    Article  MathSciNet  MATH  Google Scholar 

  28. Kellogg, R.B., Osborn, J.E.: A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal. 21(4), 397–431 (1976). https://doi.org/10.1016/0022-1236(76)90035-5

    Article  MathSciNet  MATH  Google Scholar 

  29. Dauge, M.: Stationary Stokes and Navier–Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations. SIAM J. Math. Anal. 20(1), 74–97 (1989). https://doi.org/10.1137/0520006

    Article  MathSciNet  MATH  Google Scholar 

  30. Kozlov, V.A., Maz’ya, V.G., Schwab, C.: On singularities of solutions to the Dirichlet problem of hydrodynamics near the vertex of a cone. J. Reine Angew. Math. 456, 65–97 (1994)

    MathSciNet  MATH  Google Scholar 

  31. Mitrea, M., Wright, M.: Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Astérisque (344), 241 (2012)

  32. Maz’ya, V., Rossmann, J.: \(L_p\) estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains. Math. Nachr. 280(7), 751–793 (2007). https://doi.org/10.1002/mana.200610513

    Article  MathSciNet  MATH  Google Scholar 

  33. Maz’ya, V., Rossmann, J.: Elliptic Equations in Polyhedral Domains. Mathematical Surveys and Monographs, vol. 162, p. 608. American Mathematical Society, Providence RI (2010). https://doi.org/10.1090/surv/162

    Book  MATH  Google Scholar 

  34. Rossmann, J.: Green’s matrix of the Stokes system in a convex polyhedron. Rostock. Math. Kolloq. 65, 15–28 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Mathematics, vol. 5, p. 374. Springer (1986). Theory and algorithms. https://doi.org/10.1007/978-3-642-61623-5

  36. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, p. 530. SIAM, Philadelphia PA (2002). https://doi.org/10.1137/1.9780898719208

    Book  MATH  Google Scholar 

  37. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159, p. 524. Springer, New York (2004)

    Book  Google Scholar 

  38. Guzmán, J., Salgado, A.J., Sayas, F.: A note on the Ladyženskaja-Babuška-Brezzi condition. J. Sci. Comput. 56(2), 219–229 (2013). https://doi.org/10.1007/s10915-012-9670-z

    Article  MathSciNet  MATH  Google Scholar 

  39. Nochetto, R.H., Otárola, E., Salgado, A.J.: Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications. Numer. Math. 132(1), 85–130 (2016). https://doi.org/10.1007/s00211-015-0709-6

    Article  MathSciNet  MATH  Google Scholar 

  40. Casas, E.: \(L^2\) estimates for the finite element method for the Dirichlet problem with singular data. Numer. Math. 47(4), 627–632 (1985). https://doi.org/10.1007/BF01389461

    Article  MathSciNet  MATH  Google Scholar 

  41. Drelichman, I., Durán, R.G., Ojea, I.: A weighted setting for the numerical approximation of the Poisson problem with singular sources. SIAM J. Numer. Anal. 58(1), 590–606 (2020). https://doi.org/10.1137/18M1213105

    Article  MathSciNet  MATH  Google Scholar 

  42. Bogovskiĭ, M.E.: 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)

    MathSciNet  Google Scholar 

  43. López García, F.: A decomposition technique for integrable functions with applications to the divergence problem. J. Math. Anal. Appl. 418(1), 79–99 (2014). https://doi.org/10.1016/j.jmaa.2014.03.080

    Article  MathSciNet  MATH  Google Scholar 

  44. Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics, vol. 249, 3rd edn., p. 638. Springer, Berlin (2014). https://doi.org/10.1007/978-1-4939-1194-3

    Book  MATH  Google Scholar 

  45. Roubíček, T.: Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics, vol. 153, 2nd edn., p. 476. Basel, Birkhäuser/Springer Basel AG (2013). https://doi.org/10.1007/978-3-0348-0513-1

    Book  MATH  Google Scholar 

  46. Smagorinsky, J.: General circulation experiments with the primitive equations. Monthly Weather Rev. 91(3), 99–164 (1963)

    Article  Google Scholar 

  47. Beirão da Veiga, H.: On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions. Commun. Pure Appl. Math. 58(4), 552–577 (2005). https://doi.org/10.1002/cpa.20036

    Article  MathSciNet  MATH  Google Scholar 

  48. Lesieur, M.: Turbulence in Fluids: Fourth Revised and Enlarged Edition. Fluid Mechanics and its Applications, vol. 84, 4th edn., p. 558. Springer, Berlin (2008)

    Book  Google Scholar 

  49. Sagaut, P.: Large Eddy Simulation for Incompressible Flows. Scientific Computation, p. 319. Springer (2001). https://doi.org/10.1007/978-3-662-04416-2. An introduction, With an introduction by Marcel Lesieur, Translated from the 1998 French original by the author

  50. Vreman, A.W.: The filtering analog of the variational multiscale method in large-eddy simulation. Phys. Fluids 15(8), 61–64 (2003). https://doi.org/10.1063/1.1595102

    Article  MathSciNet  MATH  Google Scholar 

  51. Dunca, A.A., Neda, M., Rebholz, L.G.: A mathematical and numerical study of a filtering-based multiscale fluid model with nonlinear eddy viscosity. Comput. Math. Appl. 66(6), 917–933 (2013). https://doi.org/10.1016/j.camwa.2013.06.013

    Article  MathSciNet  Google Scholar 

  52. Berselli, L.C., Iliescu, T., Layton, W.J.: Mathematics of Large Eddy Simulation of Turbulent Flows. Scientific Computation, p. 348. Springer, Berlin (2006)

    MATH  Google Scholar 

  53. Rappaz, J., Rochat, J.: On some weighted Stokes problems: applications on Smagorinsky models. In: Contributions to Partial Differential Equations and Applications. Comput. Methods Appl. Sci., vol. 47, pp. 395–410. Springer (2019)

  54. Hirn, A., Lanzendörfer, M., Stebel, J.: Finite element approximation of flow of fluids with shear-rate- and pressure-dependent viscosity. IMA J. Numer. Anal. 32(4), 1604–1634 (2012). https://doi.org/10.1093/imanum/drr033

    Article  MathSciNet  MATH  Google Scholar 

  55. DiBenedetto, E.: Degenerate Parabolic Equations. Universitext, p. 387. Springer, Berlin (1993). https://doi.org/10.1007/978-1-4612-0895-2

    Book  MATH  Google Scholar 

  56. Barrett, J.W., Liu, W.B.: Finite element approximation of the \(p\)-Laplacian. Math. Comp. 61(204), 523–537 (1993). https://doi.org/10.2307/2153239

    Article  MathSciNet  MATH  Google Scholar 

  57. Aimar, H., Carena, M., Durán, R., Toschi, M.: Powers of distances to lower dimensional sets as Muckenhoupt weights. Acta Math. Hungar. 143(1), 119–137 (2014). https://doi.org/10.1007/s10474-014-0389-1

    Article  MathSciNet  MATH  Google Scholar 

  58. Temam, R.: Navier–Stokes Equations, p. 408. AMS Chelsea Publishing, Providence, RI (2001). https://doi.org/10.1090/chel/343. Theory and numerical analysis, Reprint of the 1984 edition

  59. Girault, V., Nochetto, R.H., Scott, L.R.: Maximum-norm stability of the finite element Stokes projection. J. Math. Pures Appl. (9) 84(3), 279–330 (2005). https://doi.org/10.1016/j.matpur.2004.09.017

    Article  MathSciNet  MATH  Google Scholar 

  60. 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). https://doi.org/10.1007/s00211-015-0707-8

    Article  MathSciNet  MATH  Google Scholar 

  61. Maz’ya, V.G., Rossmann, J.: Schauder estimates for solutions to a mixed boundary value problem for the Stokes system in polyhedral domains. Math. Methods Appl. Sci. 29(9), 965–1017 (2006). https://doi.org/10.1002/mma.695

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Vivette Girault and Johnny Guzmán for interesting discussions and hints that led to the proof of Theorem 6. They would also like to thank Leo Rebholz for providing insight and useful references regarding the Smagorinsky model.

EO has been partially supported by CONICYT through FONDECYT project 1220156. AJS has been partially supported by NSF grants DMS-1720213 and DMS-2111228.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Abner J. Salgado.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Proof of Theorem 6

Proof of Theorem 6

The purpose of this, supplementary, section is to detail what changes, if any, are necessary to translate the results of [19, Theorem 4.1] to the case that we are interested in here. We comment that the main difference here is that, in the first equation of the definition of the Stokes projection, reference [19] employs gradients (see [19, formula (1.1)]), while we employ symmetric gradients. While in the continuous case this only amounted to a redefinition of the pressure, it rarely happens in practice that we have \({\mathrm{div}}\,{\mathbf {X}}_h \subset M_h\) and so this change of variables cannot be performed.

Let us begin then with some notation. We define

$$\begin{aligned} a(\mathbf {v},{\mathbf {w}}) = 2\mu \int _\Omega \varvec{\varepsilon }(\mathbf {v}):\varvec{\varepsilon }({\mathbf {w}}) \, \text{ d }x, \qquad b(\mathbf {v},q) = -\int _\Omega q {\mathrm{div}}\,\mathbf {v}\, \text{ d }x. \end{aligned}$$

We now realize that the heart of the matter in the proof of [19, Theorem 4.1] is the estimate provided in [19, formula (4.1)]. Thus, if we can prove

$$\begin{aligned} \Vert \varvec{\varepsilon }(\mathbf {u}_h) \Vert _{{\mathbf {L}}^2(\omega ,\Omega )} \lesssim \Vert \varvec{\varepsilon }(\mathbf {u}) \Vert _{{\mathbf {L}}^2(\omega ,\Omega )} + \Vert p \Vert _{L^2(\omega ,\Omega )}, \end{aligned}$$
(35)

the rest of the proof follows verbatim. We thus focus on the proof of (35). This is derived in several steps.

1.1 Approximate Green function

We begin the proof of (35) by defining suitable approximate Green functions. Let \(z \in \Omega \) be such that \(z \in \mathring{T}_z\), for some \(T_z \in \mathscr {T}_h\), and \({\tilde{\delta }}_z\) be a regularized Dirac delta function that satisfies the properties:

  • \({\tilde{\delta }}_z \in C_0^\infty (T_z)\);

  • \(\int _\Omega {\tilde{\delta }}_z \, \text{ d }x = 1\);

  • \(\Vert {\tilde{\delta }}_z \Vert _{L^t(T_z)} \lesssim h^{-3/t'}\), for \(t \in [1,\infty ]\);

  • \(\int _\Omega {\tilde{\delta }}_z \mathbf {v}_h \, \text{ d }x = \mathbf {v}_h(z)\) for all \(\mathbf {v}_h \in {\mathbf {X}}_h\).

The construction of such a regularized Dirac delta is presented in [59, Section 1].

Let \(z \in \Omega \) be such that \(z \in \mathring{T}_z\), for some \(T_z \in \mathscr {T}_h\), and \(i,j \in \{1,2,3\}\). We define the approximate (derivative of the) Green function as the pair \(({\mathfrak {G}},{\mathfrak {q}}) \in {\mathbf {W}}^{1,2}_0(\Omega ) \times \mathring{L}^2(\Omega )\) that solves

$$\begin{aligned} {\left\{ \begin{array}{ll} a({\mathfrak {G}},\mathbf {v}) + b(\mathbf {v},{\mathfrak {q}}) = \int _\Omega {\tilde{\delta }}_z \varvec{\varepsilon }(\mathbf {v})_{i,j} \, \text{ d }x &{} \forall \mathbf {v}\in {\mathbf {W}}^{1,2}_0(\Omega ), \\ b({\mathfrak {G}},q) = 0 &{} \forall q \in \mathring{L}^2(\Omega ). \end{array}\right. } \end{aligned}$$
(36)

We also define the Stokes projection \(({\mathfrak {G}}_h,{\mathfrak {q}}_h) \in {\mathbf {X}}_h \times M_h\) of \(({\mathfrak {G}},{\mathfrak {q}})\) via

$$\begin{aligned} {\left\{ \begin{array}{ll} a({\mathfrak {G}}_h,\mathbf {v}_h) + b(\mathbf {v}_h,{\mathfrak {q}}_h) = \int _\Omega {\tilde{\delta }}_z \varvec{\varepsilon }(\mathbf {v}_h)_{i,j} \, \text{ d }x &{} \forall \mathbf {v}_h \in {\mathbf {X}}_h, \\ b({\mathfrak {G}}_h,q_h) = 0 &{} \forall q_h \in M_h. \end{array}\right. } \end{aligned}$$
(37)

As one last ingredient, we introduce regularized distances. For \(y \in \Omega \), we define

$$\begin{aligned} \sigma _y(x) = \left( |x-y |^2 + \kappa ^2 h^2 \right) ^{1/2}, \end{aligned}$$

where \(\kappa \ge 1\) is independent of h but must satisfy that \(\kappa h \le R\) where \(R = {{\,\mathrm{diam}\,}}\Omega \). The properties of this weight are given in [59, Section 1] and [60, Section 1.7].

1.2 Reduction to weighted estimates

Having introduced the functions \(({\mathfrak {G}},{\mathfrak {q}})\) and their approximations \(({\mathfrak {G}}_h,{\mathfrak {q}}_h)\) we can proceed with the proof of (35). Upon realizing that the only property of a that is used in step 2 of the proof of [19, Theorem 4.1] is symmetry, we can follow the arguments without any change to arrive at

where we denoted \({\mathbf {E}}= {\mathfrak {G}}- {\mathfrak {G}}_h\). In conclusion, (35) holds provided we can show that the estimate

$$\begin{aligned} \sup _{z \in \Omega } \Vert \sigma _z^{\mu /2} \varvec{\varepsilon }( {\mathfrak {G}}-{\mathfrak {G}}_h) \Vert _{{\mathbf {L}}^2(\Omega )} \lesssim h^{\lambda /2} \end{aligned}$$
(38)

holds for all \(\nu \in (0,1/2)\), \(\lambda \in (0,\nu /2)\) and \(\mu = 3+\lambda \). Notice that here we are following the notation of [60] so \(\mu \) is not the viscosity.

The rest of this Appendix is dedicated to indicate what changes, if any, are necessary to prove (38).

1.3 Proof of (38)

Notice, first of all, that (38) is the analogue of [60, formula (1.46)], so we follow this reference to indicate what changes are necessary. By changing gradients to symmetric gradients, where appropriate, we can reach the analogue of [60, formula (2.3)]

$$\begin{aligned} \int _\Omega \sigma _z^\mu |\varvec{\varepsilon }({\mathbf {E}}) |^2 \, \text{ d }x&= \int _\Omega \varvec{\varepsilon }({\mathbf {E}}):\varvec{\varepsilon }(\sigma _z^\mu ({\mathfrak {G}}- P_h ({\mathfrak {G}}) )) \, \text{ d }x \\&\quad + \int _\Omega \varvec{\varepsilon }({\mathbf {E}}):\varvec{\varepsilon }({\varvec{\psi }}- {\bar{P}}_h({\varvec{\psi }}) ) \, \text{ d }x-\int _\Omega \nabla \sigma _z^\mu \cdot (\varvec{\varepsilon }({\mathbf {E}}) {\mathbf {E}}) \, \text{ d }x \\&\quad + \int _\Omega R {\mathrm{div}}\,{\bar{P}}_h({\varvec{\psi }}) \, \text{ d }x, \end{aligned}$$

where we set \(R = {\mathfrak {q}}- {\mathfrak {q}}_h\), \({\varvec{\psi }}= \sigma _z^\mu ( P_h ({\mathfrak {G}}) - {\mathfrak {G}}_h)\), and the interpolants \(P_h\) and \({\bar{P}}_h\) are described in [60, Section 1.8].

We must now derive suitable weighted bounds on the pair \(({\mathfrak {G}},{\mathfrak {q}})\), as it is done in [60, Section 2.4]. We just comment that:

  • [60, Proposition 2], which is in [59, Propostion 3.1], follows without changes, so that we obtain

    $$\begin{aligned} \Vert \sigma _z^{\mu /2-1} {\mathfrak {q}}\Vert _{L^2(\Omega )} \lesssim \Vert \sigma _z^{\mu /2-1} \varvec{\varepsilon }({\mathfrak {G}}) \Vert _{{\mathbf {L}}^2(\Omega )} + \kappa ^{\mu /2-1} h^{\lambda /2-1}. \end{aligned}$$
  • [60, Proposition 3], which is in [59, Proposition 3.2], follows with little changes to obtain

    $$\begin{aligned} \Vert \sigma _z^{\mu /2-1} \varvec{\varepsilon }({\mathfrak {G}}) \Vert _{{\mathbf {L}}^2(\Omega )}^{2}&\le \Vert \sigma _z^{\mu /2-2} {\mathfrak {G}}\Vert _{{\mathbf {L}}^2(\Omega )} \left( c_1 \kappa ^{\mu /2} h^{\lambda /2-1} \right. \\&\quad + \left. c_2\Vert \sigma _z^{\mu /2-1} \varvec{\varepsilon }({\mathfrak {G}}) \Vert _{{\mathbf {L}}^2(\Omega )} \right) . \end{aligned}$$
  • [60, Theorem 5], follows without changes, so that we obtain the analogue of [60, Corollary 1]:

    $$\begin{aligned} \Vert \sigma _z^{\mu /2-1} \varvec{\varepsilon }({\mathfrak {G}}) \Vert _{{\mathbf {L}}^2(\Omega )} + \Vert \sigma _z^{\mu /2-1} {\mathfrak {q}}\Vert _{L^2(\Omega )} \lesssim \kappa ^{\mu /4}h^{\lambda /2-1}. \end{aligned}$$
  • To obtain the regularity estimates of [60, Theorem 6], we follow the arguments given in [59, Theorem 3.6] and realize that we must compute the effect of the Stokes operator on \((\sigma _z^{\mu /2}{\mathfrak {G}},\sigma _z^{\mu /2}{\mathfrak {q}})\). After elementary computations, one realizes that

    $$\begin{aligned} -{\mathrm{div}}\,(\varvec{\varepsilon }(\sigma _z^{\mu /2}{\mathfrak {G}})) + \nabla (\sigma _z^{\mu /2}{\mathfrak {q}}) = \sigma _z^{\mu /2}\left[ -{\mathrm{div}}\,(\varvec{\varepsilon }({\mathfrak {G}})) + \nabla {\mathfrak {q}}\right] + {\varvec{\mathcal {F}}}, \end{aligned}$$

    where \({\varvec{\mathcal {F}}}\) depends on \({\mathfrak {G}}\), \(\nabla {\mathfrak {G}}\), \(\nabla \sigma _z^{\mu /2}\), and \({\mathfrak {q}}\). The important point is that \({\varvec{\mathcal {F}}}\in {\mathbf {L}}^2(\Omega )\). Using the regularity results for the Stokes operator of Proposition 2, we conclude then that the right hand side of the expression above is an element of \({\mathbf {L}}^2(\Omega )\). In addition, we have that

    $$\begin{aligned} {\mathrm{div}}\,(\sigma _z^{\mu /2}{\mathfrak {G}}) = \nabla \sigma _z^{\mu /2} \cdot {\mathfrak {G}}\in W^{1,2}_0(\Omega ) \cap \mathring{L}^2(\Omega ). \end{aligned}$$

    In conclusion, upon invoking Proposition 2 once again, we have that

    $$\begin{aligned} \Vert \sigma _z^{\mu /2} D^2{\mathfrak {G}}\Vert _{{\mathbf {L}}^2(\Omega )} + \Vert \sigma _z^{\mu /2} \nabla {\mathfrak {q}}\Vert _{{\mathbf {L}}^2(\Omega )} \lesssim \kappa ^{\mu /2}h^{\lambda /2-1} \end{aligned}$$

    and [60, Theorem 7] follows without changes.

The discussion of [60, Section 3] is about finite element spaces, and so it does not need any changes.

At this point we have set the stage to carry out the bootstrap procedure of [60, Section 4]. To carry out the duality argument in the proof of Theorem 9, we must introduce the pair \(({\varvec{\varphi }},s) \in {\mathbf {W}}^{1,2}_0(\Omega ) \times \mathring{L}^2(\Omega )\) that solves

$$\begin{aligned} -{\mathrm{div}}\,(\varvec{\varepsilon }({\varvec{\varphi }})) + \nabla s = \sigma _z^{\mu +\epsilon -2}({\mathfrak {G}}-{\mathfrak {G}}_h), \quad {\mathrm{div}}\,{\varvec{\varphi }}= 0, \ \text { in } \Omega , \quad {\varvec{\varphi }}= 0, \ \text { on } \partial \Omega . \end{aligned}$$

The redefinition of the pressure indicated in Remark 4 allows us to conclude, owing to [61], that there is \(\alpha \in (0,1)\) that depends on \(\Omega \) and for which we have \(({\varvec{\varphi }},s) \in {\mathbf {C}}^{1,\alpha }({\bar{\Omega }}) \times C^{0,\alpha }({\bar{\Omega }})\) with an estimate similar to [60, estimate (4.4)]. Upon replacing gradients by symmetric gradients in [60, formulas (4.5), (4.6), and (4.7)] we arrive at the analogue of [60, estimate (4.8)]:

$$\begin{aligned}&\left\| \sigma _z^{\frac{1}{2}(\mu +\epsilon )-1}({\mathfrak {G}}- {\mathfrak {G}}_h) \right\| _{{\mathbf {L}}^2(\Omega )}^2 \\&\quad \le \Vert \sigma _z^{-\mu /2} \varvec{\varepsilon }({\varvec{\varphi }}- {\bar{P}}_h({\varvec{\varphi }})) \Vert _{{\mathbf {L}}^2(\Omega )} \Vert \sigma _z^{\mu /2}\varvec{\varepsilon }({\mathfrak {G}}- {\mathfrak {G}}_h ) \Vert _{{\mathbf {L}}^2(\Omega )} \\&\qquad + \Vert \sigma _z^{-\mu /2} {\mathrm{div}}\,({\varvec{\varphi }}- {\bar{P}}_h({\varvec{\varphi }})) \Vert _{L^2(\Omega )} \Vert \sigma _z^{\mu /2} ({\mathfrak {q}}- r_h({\mathfrak {q}})) \Vert _{L^2(\Omega )} \\&\qquad - \int _\Omega (s - {\bar{r}}_h(s) ) {\mathrm{div}}\,({\mathfrak {G}}-{\mathfrak {G}}_h) \, \text{ d }x = \mathrm {I} + \mathrm {II} + \mathrm {III}, \end{aligned}$$

where \(\epsilon \in (0,1)\) is as in [60, (4.8)], \(r_h\) and \(\bar{r}_h\) are the interpolants described in [60, Section 1.8]. Let us now show the slight departures from the argument in [60] with some detail. As in [60], the regularity of \(({\varvec{\varphi }},s)\) implies that

$$\begin{aligned}&\Vert \sigma _z^{-\mu /2} \varvec{\varepsilon }({\varvec{\varphi }}- {\bar{P}}_h({\varvec{\varphi }})) \Vert _{{\mathbf {L}}^2(\Omega )} + \Vert \sigma _z^{-\mu /2} {\mathrm{div}}\,({\varvec{\varphi }}- {\bar{P}}_h({\varvec{\varphi }})) \Vert _{L^2(\Omega )} \\&\quad + \Vert \sigma _z^{-\mu /2}(s - {\bar{r}}_h(s)) \Vert _{L^2(\Omega )} \lesssim h^\alpha (\kappa h)^{-\lambda /2} \left\| \sigma _z^{\mu +\epsilon -2}({\mathfrak {G}}- {\mathfrak {G}}_h) \right\| _{{\mathbf {L}}^r(\Omega )}, \end{aligned}$$

where \(\alpha = 1-3/r\). The estimate on terms \(\mathrm {I}\) and \(\mathrm {II}\) then proceeds as in [60]. The estimate of \(\mathrm {III}\) is obtained by noticing that, for any matrix M, \(|{{\,\mathrm{tr}\,}}M |\lesssim |M |\). In addition, for any vector field \(\mathbf {v}\), we have \({{\,\mathrm{tr}\,}}\varvec{\varepsilon }(\mathbf {v}) = {\mathrm{div}}\,\mathbf {v}\). These observations allow us to write

$$\begin{aligned} \mathrm {III}&\le \Vert \sigma _z^{-\mu /2}(s - {\bar{r}}_h(s)) \Vert _{L^2(\Omega )} \Vert \sigma _z^{\mu /2} {\mathrm{div}}\,({\mathfrak {G}}- {\mathfrak {G}}_h)\Vert _{L^2(\Omega )} \\&\quad \lesssim \Vert \sigma _z^{-\mu /2}(s - {\bar{r}}_h(s)) \Vert _{L^2(\Omega )} \Vert \sigma _z^{\mu /2} \varvec{\varepsilon }({\mathfrak {G}}- {\mathfrak {G}}_h)\Vert _{{\mathbf {L}}^2(\Omega )}, \end{aligned}$$

and the estimate on \(\mathrm {III}\) now proceeds as in [60].

We have thus arrived at [60, (4.10)], where one can invoke Korn’s inequality (4) with \(q=2\) and \(\omega \equiv 1\), to replace the gradient by a symmetric gradient. We can keep replacing gradients by \(\varvec{\varepsilon }\) to arrive at the conclusion of [60, Theorem 9].

Once [60, Theorem 9] holds, [60, Corollaries 2 and 3] are a simple exercise and thus we obtain

$$\begin{aligned} \Vert \sigma _z^{\mu /2-1} ({\mathfrak {G}}- {\mathfrak {G}}_h) \Vert _{{\mathbf {L}}^2(\Omega )}^2 \lesssim \frac{1}{\kappa ^\alpha } \Vert \sigma _z^{\mu /2}\varvec{\varepsilon }({\mathfrak {G}}- {\mathfrak {G}}_h) \Vert _{{\mathbf {L}}^2(\Omega )}^2 + \kappa ^{\mu -\alpha } h^\lambda , \end{aligned}$$

and

$$\begin{aligned} \left\| \sigma _z^{\frac{1}{2}(\mu +\epsilon )-1}({\mathfrak {G}}- {\mathfrak {G}}_h) \right\| _{{\mathbf {L}}^2(\Omega )}^2 \le \frac{(\kappa h)^\epsilon }{\kappa ^\alpha } \left( C_\kappa \Vert \sigma _z^{\mu /2} \varvec{\varepsilon }({\mathfrak {G}}- {\mathfrak {G}}_h) \Vert _{{\mathbf {L}}^2(\Omega )}^2 + \kappa ^\mu h^\lambda \right) , \end{aligned}$$

where \(C_\kappa := 1 + \tfrac{1}{\kappa ^\alpha }\).

The estimates on the pressure term [60, Section 5] hold with little or no modification. In [60, Lemmas 7 and 8] one only needs to replace gradients with symmetric gradients. The same is true for [60, Theorem 10], which is [59, Theorem 4.2]. Then, [60, Proposition 10 and Theorem 11] need no changes.

This, finally, brings us to [60, Section 6], where [60, Theorem 12] proceeds without changes, proves (38), and concludes our argument.

As a final remark, we comment that in this new setting [60, Theorem 13 and Corollary 4] also follow without changes. The pressure estimates of [60, Section 6.2] only require to change [60, (6.6)] accordingly.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Otárola, E., Salgado, A.J. On the analysis and approximation of some models of fluids over weighted spaces on convex polyhedra. Numer. Math. 151, 185–218 (2022). https://doi.org/10.1007/s00211-022-01272-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-022-01272-5

Mathematics Subject Classification

Navigation