A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows. (English) Zbl 1228.76084

Summary: In this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The methods are based on a non-standard mixed approach in which the velocity, the pressure, and the pseudostress are the original unknowns. However, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown, which yields a twofold saddle point operator equation as the resulting dual-mixed variational formulation. In addition, a suitable augmented version of the latter showing a single saddle point structure is also considered. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, we show that Raviart-Thomas elements of order \(k\leq 0\) for the pseudostress, and piecewise polynomials of degree \(k\) for the velocity and its gradient, ensure the well-posedness of the associated Galerkin schemes. Moreover, we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation. Then, we derive a reliable and efficient residual-based a posteriori error estimator for each scheme. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithms.


76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
65N15 Error bounds for boundary value problems involving PDEs
Full Text: DOI


[1] Arnold, D.N.; Brezzi, F.; Douglas, J., PEERS: a new mixed finite element method for plane elasticity, Japan J. appl. math., 1, 347-367, (1984) · Zbl 0633.73074
[2] Arnold, D.N.; Douglas, J.; Gupta, Ch.P., A family of higher order mixed finite element methods for plane elasticity, Numerische Mathematik, 45, 1-22, (1984) · Zbl 0558.73066
[3] Baranger, J.; Najib, K.; Sandri, D., Numerical analysis of a three-fields model for a quasi-Newtonian flow, Comput. methods appl. mech. engrg., 109, 3-4, 281-292, (1993) · Zbl 0844.76004
[4] Bochev, P.B.; Gunzburger, M.D., Least-squares methods for the velocity-pressure-stress formulation of the Stokes equations, Comput. methods appl. mech. engrg., 126, 3-4, 267-287, (1995) · Zbl 1067.76562
[5] Bochev, P.B.; Gunzburger, M.D., Finite element methods of least-squares type, SIAM rev., 40, 4, 789-837, (1998) · Zbl 0914.65108
[6] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer Verlag · Zbl 0788.73002
[7] Cai, Z.; Lee, B.; Wang, P., Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems, SIAM J. numer. anal., 42, 2, 843-859, (2004) · Zbl 1159.76347
[8] Cai, Z.; Manteuffel, T.A.; McCormick, S.F., First-order system least squares for the Stokes equations, with application to linear elasticity, SIAM J. numer. anal., 34, 5, 1727-1741, (1997) · Zbl 0901.76052
[9] Cai, Z.; Starke, G., Least-squares methods for linear elasticity, SIAM J. numer. anal., 42, 2, 826-842, (2004) · Zbl 1159.74419
[10] Cai, Z.; Tong, Ch.; Vassilevski, P.S.; Wang, Ch., Mixed finite element methods for incompressible flow: stationary Stokes equations, Numer. methods partial differ. equat., 26, 4, 957-978, (2009) · Zbl 1267.76059
[11] Cai, Z.; Wang, Y., A multigrid method for the pseudostress formulation of Stokes problems, SIAM J. sci. comput., 29, 5, 2078-2095, (2007) · Zbl 1182.76896
[12] Carstensen, C., A posteriori error estimate for the mixed finite element method, Math. comput., 66, 218, 465-476, (1997) · Zbl 0864.65068
[13] Chang, C.L., A mixed finite element method for the Stokes problem: an acceleration-pressure formulation, Appl. math. comput., 36, 2, 135-146, (1990) · Zbl 0702.76075
[14] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland · Zbl 0445.73043
[15] Clément, P., Approximation by finite element functions using local regularisation, RAIRO modélisation mathématique et analyse numérique, 9, 77-84, (1975) · Zbl 0368.65008
[16] Ervin, V.J.; Howell, J.S.; Stanculescu, I., A dual-mixed approximation method for a three-field model of a nonlinear generalized Stokes problem, Comput. methods appl. mech. engrg., 197, 33-40, 2886-2900, (2008) · Zbl 1194.76114
[17] Figueroa, L.; Gatica, G.N.; Heuer, N., A priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows, Comput. methods appl. mech. engrg., 198, 2, 280-291, (2008) · Zbl 1194.76115
[18] Figueroa, L.; Gatica, G.N.; Márquez, A., Augmented mixed finite element methods for the stationary Stokes equations, SIAM J. sci. comput., 31, 2, 1082-1119, (2008) · Zbl 1251.74032
[19] Gatica, G.N., Solvability and Galerkin approximations of a class of nonlinear operator equations, Zeitschrif für analysis und ihre anwendungen, 21, 3, 761-781, (2002) · Zbl 1024.65044
[20] Gatica, G.N., An application of babuška – brezzi theory to a class of variational problems, Applicable anal., 75, 297-303, (2000) · Zbl 1021.65030
[21] Gatica, G.N., Analysis of a new augmented mixed finite element method for linear elasticity allowing \(\mathbb{RT}_0 - \mathbb{P}_1 - \mathbb{P}_0\) approximations, ESAIM math. model. numer. anal., 40, 1, 1-28, (2006)
[22] Gatica, G.N., An augmented mixed finite element method for linear elasticity with non-homogeneous Dirichlet conditions, Electron. trans. numer. anal., 26, 421-438, (2007) · Zbl 1170.74049
[23] Gatica, G.N.; González, M.; Meddahi, S., A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. I: a priori error analysis, Comput. methods appl. mech. engrg., 193, 9-11, 881-892, (2004) · Zbl 1053.76037
[24] Gatica, G.N.; Heuer, N.; Meddahi, S., On the numerical analysis of nonlinear two-fold saddle point problems, IMA J. numer. anal., 23, 2, 301-330, (2003) · Zbl 1028.65128
[25] Gatica, G.N.; Márquez, A.; Meddahi, S., An augmented mixed finite element method for 3D linear elasticity problems, J. comput. appl. math., 231, 2, 526-540, (2009) · Zbl 1167.74042
[26] Gatica, G.N.; Márquez, A.; Sánchez, M.A., Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations, Comput. methods appl. mech. engrg., 199, 17-20, 1064-1079, (2010) · Zbl 1227.76030
[27] Gatica, G.N.; Meddahi, S., A dual-dual mixed formulation for nonlinear exterior transmission problems, Math. comput., 70, 236, 1461-1480, (2001) · Zbl 0980.65132
[28] Girault, V.; Raviart, P.-A., Finite element methods for navier – stokes equations. theory and algorithms, Springer series in computational mathematics, vol. 5, (1986), Springer-Verlag
[29] Hiptmair, R., Finite elements in computational electromagnetism, Acta numerica, 11, 237-339, (2002) · Zbl 1123.78320
[30] Howell, J.S., Dual-mixed finite element approximation of Stokes and nonlinear Stokes problems using trace-free velocity gradients, J. comput. appl. math., 231, 2, 780-792, (2009) · Zbl 1167.76021
[31] Ladyzhenskaya, O., New equations for the description of the viscous incompressible fluids and solvability in the large for the boundary value problems of them, in: Boundary Value Problems of Mathematical Physics V, Providence, RI: AMS, 1970.
[32] Larsson, F.; Diez, P.; Huerta, A., A flux-free a posteriori error estimator for the incompressible Stokes problem using a mixed FE formulation, Comput. methods appl. mech. engrg., 199, 37-40, 2383-2402, (2010) · Zbl 1231.76154
[33] Loula, A.F.D.; Guerreiro, J.N.C., Finite element analysis of nonlinear creeping flows, Comput. methods appl. mech. engrg., 79, 1, 87-109, (1990) · Zbl 0716.73091
[34] Quarteroni, A.; Valli, A., Numerical approximation of partial differential equations, Springer series in computational mathematics, vol. 23, (1994), Springer-Verlag Berlin Heidelberg · Zbl 0852.76051
[35] Roberts, J.E.; Thomas, J.M., Mixed and hybrid methods, () · Zbl 0875.65090
[36] Sandri, D., Sur l’approximation numérique des écoulements quasi-newtoniens dont la viscosité suit la loi puissance ou la loi de carreau, Math. model. numer. anal., 27, 2, 131-155, (1993) · Zbl 0764.76039
[37] Scheurer, B., Existence et approximation de point selles pour certain problemes non linéaires, R.A.I.R.O. analyse numérique, 11, 4, 369-400, (1977) · Zbl 0371.65025
[38] Verfürth, R., A review of A posteriori error estimation and adaptive mesh-refinement techniques, (1996), Wiley-Teubner Chichester · Zbl 0853.65108
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