Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology

Kreuzer, Christian; Süli, Endre

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 5, pp. 1333-1369.

We develop the a posteriori error analysis of finite element approximations to implicit power-law-like models for viscous incompressible fluids in $d$ space dimensions, $d\in \{2,3\}$. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi-valued, maximal monotone $r$-graph, with $\frac{2d}{d+1}<r<\infty$. We establish upper and lower bounds on the finite element residual, as well as the local stability of the error bound. We then consider an adaptive finite element approximation of the problem, and, under suitable assumptions, we show the weak convergence of the adaptive algorithm to a weak solution of the boundary-value problem. The argument is based on a variety of weak compactness techniques, including Chacon’s biting lemma and a finite element counterpart of the Acerbi–Fusco Lipschitz truncation of Sobolev functions, introduced by [L. Diening, C. Kreuzer and E. Süli, SIAM J. Numer. Anal. 51 (2013) 984–1015].

E. Acerbi and N. Fusco, An approximation lemma for $W^{1,p}$ functions, Material instabilities in continuum mechanics (Edinburgh, 1985–1986). Oxford Univ. Press, New York (1988) 1–5. | MR | Zbl

M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis. Wiley-Interscience, New York (2000). | MR | Zbl

C.D. Aliprantis and K.C. Border, Infinite dimensional analysis, A hitchhiker’s guide, 3rd edition. Springer, Berlin (2006). | MR | Zbl

J.-P. Aubin and H. Frankowska, Set-valued analysis, Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA (2009), Reprint of the 1990 edition. | MR | Zbl

E. Bänsch, Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Engrg. 3 (1991) 181–191. | DOI | MR | Zbl

L. Belenki, L. Berselli, L. Diening and M. Rzŭˇička, On the finite element approximation of p-stokes systems. SIAM J. Numer. Anal. 50 (2012) 373–397. | DOI | MR | Zbl

L. Belenki, L. Diening and C. Kreuzer, Quasi-optimality of an adaptive finite element method for the $p$-Laplacian equation. IMA J. Numer. Anal. 32 (2012) 484–510. | DOI | MR | Zbl

S. Berrone and E. Süli, Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows. IMA J. Numer. Anal. 28 (2008) 382–421. | DOI | MR | Zbl

M.E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR 248 (1979) 1037–1040. | MR | Zbl

D. Breit, L. Diening and M. Fuchs, Solenoidal Lipschitz truncation and applications in fluid mechanics. J. Differ. Eq. 253 (2012) 1910–1942. | DOI | MR | Zbl

D. Breit, L. Diening and S. Schwarzacher, Solenoidal Lipschitz truncation for parabolic PDEs. Math. Models Methods Appl. Sci. 23 (2013) 2671–2700. | DOI | MR | Zbl

F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Vol. 15 of Springer Series in Computational Mathematics (1991). | MR | Zbl

J.K. Brooks and R.V. Chacon, Continuity and compactness of measures. Adv. Math. 37 (1980) 16–26. | DOI | MR | Zbl

M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On steady flows of incompressible fluids with implicit power-law-like rheology. Adv. Calc. Var. 2 (2009) 109–136. | DOI | MR | Zbl

M. Bulíček, P. Gwiazda, J. Málek, K.R. Rajagopal and A. Świerczewska-Gwiazda, On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph. Mathematical Aspects of Fluid Mechanics, London Mathematical Society Lecture Note Series (No. 402). Cambridge University Press (2012) 23–51. | MR | Zbl

M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska-Gwiazda, On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44 (2012) 2756–2801. | DOI | MR | Zbl

L. Diening and M. Rŭžička, Interpolation operators in Orlicz–Sobolev spaces. Numer. Math. 107 (2007) 107–129. | DOI | MR | Zbl

L. Diening and C. Kreuzer, Convergence of an adaptive finite element method for the $p$-Laplacian equation. SIAM J. Numer. Anal. 46 (2008) 614–638. | DOI | MR | Zbl

L. Diening, J. Málek, and M. Steinhauer, On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM: COCV 14 (2008) 211–232. | Numdam | MR | Zbl

L. Diening, M. Rŭžička and K. Schumacher, A decomposition technique for John domains. Ann. Acad. Sci. Fenn. Math. 35 (2010) 87–114. | DOI | MR | Zbl

L. Diening, C. Kreuzer and E. Süli, Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. Prpeprint (2012). | arXiv | MR | Zbl

L. Diening, C. Kreuzer and E. Süli, Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal. 51 (2013) 984–1015. | DOI | MR | Zbl

G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics. [Translated from the French by C.W. John. Grundlehren der Mathematischen Wissenschaften, 219.] Springer-Verlag, Berlin (1976). | MR | Zbl

V. Girault and J.-L. Lions, Two-grid finite-element schemes for the steady Navier–Stokes problem in polyhedra. Port. Math. (N.S.) 58 (2001) 25–57. | MR | Zbl

V. Girault and L.R. Scott, A quasi-local interpolation operator preserving the discrete divergence. Calcolo 40 (2003) 1–19. | DOI | MR | Zbl

P. Gwiazda and A. Zatorska-Goldstein, On elliptic and parabolic systems with $x$-dependent multivalued graphs. Math. Methods Appl. Sci. 30 (2007) 213–236. | DOI | MR | Zbl

P. Gwiazda, J. Málek and A. Świerczewska, On flows of an incompressible fluid with discontinuous power-law-like rheology. Comput. Math. Appl. 53 (2007) 531–546. | DOI | MR | Zbl

J. Hron, J. Málek, J. Stebel and K. Touška, A novel view of computations of steady flows of Bingham and Herschel–Bulkley fluids using implicit constitutive relations. Proc. of the 26th Nordic Seminar on Computational Mechanics, Oslo, 23–25 October 2013, edited by A. Logg, K.-A. Mardal and A. Massing. Center for Biomedical Computing, Simula Research Laboratory, Oslo (2013) 217–219.

I. Kossaczký, A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math. 55 (1994) 275–288. | DOI | MR | Zbl

C. Kreuzer and E. Süli, Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. Preprint (2015). | arXiv | Numdam | MR | Zbl

W. Liu and N. Yan, Quasi-norm local error estimators for $p$-Laplacian.. SIAM J. Numer. Anal. 39 (2001) 100–127. | DOI | MR | Zbl

J. Málek, Mathematical properties of the flows of incompressible fluids with pressure and shear rate dependent viscosities. D.Sc. thesis, Academy of Sciences of the Czech Republic, Prague (2007).

J. Málek, Mathematical properties of flows of incompressible power-law-like fluids that are described by implicit constitutive relations. Electron. Trans. Numer. Anal. 31 (2008) 110–125. | MR | Zbl

P. Morin, K.G. Siebert, A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. 18 (2008) 707–737. | DOI | MR | Zbl

K.R. Rajagopal, On implicit constitutive theories. Appl. Math., Praha 48 (2003) 279–319. | DOI | MR | Zbl

K.R. Rajagopal, On implicit constitutive theories for fluids. J. Fluid Mech. 550 (2006) 243–249. | DOI | MR | Zbl

K.R. Rajagopal and A.R. Srinivasa, On the thermodynamics of fluids defined by implicit constitutive relations. Z. Angew. Math. Phys. 59 (2008) 715–729. | DOI | MR | Zbl

K.G. Siebert, A convergence proof for adaptive finite elements without lower bound. IMA J. Numer. Anal. 31 (2011) 947–970. | DOI | MR | Zbl

A. Schmidt and K.G. Siebert, Design of adaptive finite element software. The finite element toolbox ALBERTA. Vol. 42 of Lect. Notes Comput. Sci. Engrg. Springer (2005). | MR | Zbl

R. Stevenson, The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77 (2008) 227–241. | DOI | MR | Zbl

R. Temam, Navier–Stokes equations. Theory and numerical analysis. Vol. 2 of Studies in Mathematics and its Applications, 3rd edition. North-Holland, Amsterdam, New York, Oxford (1984). | Zbl

R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Adv. Numer. Math. John Wiley, Chichester, UK (1996). | Zbl

R. Verfürth, A posteriori error estimation techniques for finite element methods. Numerical Mathematics and Scientific Computation. The Clarendon Press, Oxford University Press, New York (2013). | MR | Zbl