Startseite Error analysis of finite element and finite volume methods for some viscoelastic fluids
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Error analysis of finite element and finite volume methods for some viscoelastic fluids

  • Mária Lukáčová-Medvid’ová EMAIL logo , Hana Mizerová , Bangwei She und Jan Stebel
Veröffentlicht/Copyright: 8. Juni 2016

Abstract

We present the error analysis of a particular Oldroyd-B type model with the limiting Weissenberg number going to infinity. Assuming a suitable regularity of the exact solution we study the error estimates of a standard finite element method and of a combined finite element/finite volume method. Our theoretical result shows first order convergence of the finite element method and the error of the order 𝓞(h3/4) for the finite element/finite volume method. These error estimates are compared and confirmed by the numerical experiments.

Acknowledgment

B.S. and H.M. would like to thank Profs. Tabata and Notsu (Waseda University, Tokyo) for fruitful discussions on the topic.

References

[1] J.W. Barrett and S. Boyaval, Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21(2011), 1783–1837.10.1142/S0218202511005581Suche in Google Scholar

[2] S. Boyaval, T. Lelièvre, and C. Mangoubi, Free-energy-dissipative schemes for the Oldroyd-B model, Math. Modelling Num. Anal. 43(2009), 523–561.10.1051/m2an/2009008Suche in Google Scholar

[3] S. C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15, Springer, 2008.10.1007/978-0-387-75934-0Suche in Google Scholar

[4] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Comp. Math., Springer, 1991.10.1007/978-1-4612-3172-1Suche in Google Scholar

[5] X. Chen, H. Marschall, M. Schäfer, and D. Bothe, A comparison of stabilisation approaches for finite-volume simulation of viscoelastic fluid flow, Int. J. Comput. Fluid Dyn. 27(2013), 229–250.10.1080/10618562.2013.829916Suche in Google Scholar

[6] Y. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions, Comm. Partial Differ. Equ. 31(2006), 1793–1810.10.1080/03605300600858960Suche in Google Scholar

[7] P. Constantin and M. Kliegl, Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress, Arch. Ration. Mech. Anal. 206(2012), 725–740.10.1007/s00205-012-0537-0Suche in Google Scholar

[8] P. Constantin and W. Sun, Remarks on Oldroyd-B and related complex fluid models, Commun. Math. Sci. 10(2012), 33–73.10.4310/CMS.2012.v10.n1.a3Suche in Google Scholar

[9] B.J. Daly, F.R. Harlow, J.P. Shannon, and J.E. Welch, The MAC method, Technical report LA-3425, Los Alamos Scientific Laboratory, University of California (1965).Suche in Google Scholar

[10] H. Damanik, J. Hron, A. Ouazzi, and S. Turek, A monolithic FEM approach for the log-conformation reformulation (LCR) of viscoelastic flow problems, J. Non-Newton. Fluid Mech. 165(2010), 1105–1113.10.1016/j.jnnfm.2010.05.008Suche in Google Scholar

[11] V. Dolejší, M. Feistauer, and C. Schwab, A finite volume discontinuous Galerkin scheme for nonlinear convection–diffusion problems, Calcolo39(2002), 1–40.10.1007/s100920200000Suche in Google Scholar

[12] R. Eymard, R. Herbin, and J. C. Latché, On a stabilized colocated finite volume scheme for the Stokes problem, M2AN Math. Model. Numer. Anal. 40(2006), 501–527.10.1051/m2an:2006024Suche in Google Scholar

[13] R. Fattal and R. Kupferman, Time-dependent simulation of viscoelastic flows at high Weissenberg number using the logconformation representation, J. Non-Newton. Fluid Mech. 126(2005), 23–37.10.1016/j.jnnfm.2004.12.003Suche in Google Scholar

[14] M. Feistauer, M. Krížek, and V. Sobotíková, An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type, East-West J. Numer. Math. 1(1993), 267–285.Suche in Google Scholar

[15] E. Fernández-Cara, E. Guillén, and R.R. Ortega, Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, in: Handbook of Numerical Analysis, Vol. VIII, pp. 543–661, North-Holland, Amsterdam, 2002.10.1016/S1570-8659(02)08005-5Suche in Google Scholar

[16] V. Girault and H. Lopez, Finite-element error estimates for the MAC scheme, IMA J. Numer. Anal. 16(1996), 247–379.10.1093/imanum/16.3.347Suche in Google Scholar

[17] C. Guillope and J.C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Modélisation Math. Anal. Numér. 24(1990), 369–401.10.1051/m2an/1990240303691Suche in Google Scholar

[18] R. Herbin, J.-C. Latché, and K. Mallem, Convergence of the MAC scheme for the steady-state incompressible Navier–Stokes equations on non-uniform grids, in: Finite volumes for complex applications, Vol. VII. Methods and theoretical aspects, Springer Proc. Math. Stat. 77, pp. 343–351, Springer, Cham, 2014.10.1007/978-3-319-05684-5_33Suche in Google Scholar

[19] M. Hieber, Y. Naito, and Y. Shibata, Global existence results for Oldroyd-B fluids in exterior domains, J. Differ. Equ. 252(2012), 2617–2629.10.1016/j.jde.2011.09.001Suche in Google Scholar

[20] G. Kanschat, Divergence-free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme, Int. J. Numer. Methods Fluids56(2008), 941–950.10.1002/fld.1566Suche in Google Scholar

[21] O. Kreml and M. Pokorný, On the local strong solutions for a system describing the flow of a viscoelastic fluid, Banach Center Publ. 86(2009), 195–206.10.4064/bc86-0-12Suche in Google Scholar

[22] V.I. Lebedev, Difference analogues of orthogonal decompositions, basic differential operators and som boundary problems of mathematical physics. I, USSR Comp. Math. Math. Phys. 4(1964), 69–92.10.1016/0041-5553(64)90240-XSuche in Google Scholar

[23] J. Li and S. Sun, The superconvergence phenomenon and proof of the MAC scheme for the Stokes equations on nonuniform rectangular meshes, J. Sci. Comput. (2014), doi:10.1007/s10915-014-9963-5.Suche in Google Scholar

[24] F.-H. Lin, C. Liu, and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math. 58(2005), 1437–1471.10.1002/cpa.20074Suche in Google Scholar

[25] P.-L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math., Ser. B21(2000), 131–146.10.1142/S0252959900000170Suche in Google Scholar

[26] R. A. Nicolaides, Analysis and convergence of the MAC scheme. I. The linear problem, SIAM J. Numer. Anal. 29(1992), 1579–1591.10.1137/0729091Suche in Google Scholar

[27] R. A. Nicolaides and X. Wu, Numerical solution of the Hamel problem by a covolume method, in: Advances in Computational Fluid Dynamics (Eds. W.G. Habashi and M. Hafez), pp. 397–422, Gordon and Breach, 1995.Suche in Google Scholar

[28] R. A. Nicolaides and X. Wu, Analysis and convergence of the MAC scheme. II. Navier–Stokes equations, Math. Comp. 65(1996), 29–44.10.1090/S0025-5718-96-00665-5Suche in Google Scholar

Received: 2014-8-8
Revised: 2015-2-1
Accepted: 2015-2-3
Published Online: 2016-6-8
Published in Print: 2016-6-1

© 2016 by Walter de Gruyter Berlin/Boston

Heruntergeladen am 26.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jnma-2014-0057/pdf
Button zum nach oben scrollen