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A hybrid high-order method for a class of quasi-Newtonian Stokes equations on general meshes. (English) Zbl 1433.76098

Summary: In this paper, we introduce a hybrid high-order (HHO) discrete scheme for numerically solving a class of incompressible quasi-Newtonian Stokes equations in \(\mathbb{R}^2\). The presented HHO method depends on hybrid discrete velocity unknowns at cells and edges, and pressure unknowns at cells. Benefiting from the hybridization of unknowns, the computation cost can be reduced by the technique of static condensation and the solvability of the static condensation algebra system is proved. Furthermore, we study the HHO scheme by polynomials of arbitrary degrees \(k\) \((k \geq 1)\) on the general meshes and geometries. The unique solvability of the discrete scheme is proved. Additionally, the optimal a priori error estimates for the velocity gradient and pressure approximations are obtained. Finally, we provide several numerical results to verify the good performance of the proposed HHO scheme and confirm the optimal approximation properties on a variety of meshes and geometries.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows

Software:

PolyMesher; iFEM
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Full Text: DOI

References:

[1] Saramito, P., Complex Fluids. Modeling and Algorithms. Complex Fluids. Modeling and Algorithms, Mathématiques & Applications (Berlin) [Mathematics & Applications], 79 (2016), Springer: Springer Cham · Zbl 1361.76001
[2] Loula, A. F.D.; Guerreiro, J. N.C., Finite element analysis of nonlinear creeping flows, Comput. Methods Appl. Mech. Eng., 79, 1, 87-109 (1990) · Zbl 0716.73091
[3] Baranger, J.; Najib, K.; Sandri, D., Numerical analysis of a three-fields model for a quasi-Newtonian flow, Comput. Methods Appl. Mech. Eng., 109, 3-4, 281-292 (1993) · Zbl 0844.76004
[4] 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) · Zbl 0811.76036
[5] Manouzi, H.; Farhloul, M., Mixed finite element analysis of a non-linear three-fields Stokes model, IMA J. Numer. Anal., 21, 1, 143-164 (2001) · Zbl 0971.76049
[6] 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) · Zbl 0796.76049
[7] Du, Q.; Gunzburger, M. D., Finite-element approximations of a Ladyzhenskaya model for stationary incompressible viscous flow, SIAM J. Numer. Anal., 27, 1, 1-19 (1990) · Zbl 0697.76046
[8] Bao, W.; Barrett, J. W., A priori and a posteriori error bounds for a nonconforming linear finite element approximation of a non-Newtonian flow, RAIRO Modél. Math. Anal. Numér., 32, 7, 843-858 (1998) · Zbl 0912.76025
[9] Gatica, G. N.; González, M.a.; Meddahi, S., A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. II. A posteriori error analysis, Comput. Methods Appl. Mech. Eng., 193, 9-11, 893-911 (2004) · Zbl 1053.76038
[10] Gatica, G. N.; González, M.a.; 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. Eng., 193, 9-11, 881-892 (2004) · Zbl 1053.76037
[11] 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. Eng., 197, 33-40, 2886-2900 (2008) · Zbl 1194.76114
[12] 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
[13] Congreve, S.; Houston, P.; Süli, E.; Wihler, T. P., Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: strongly monotone quasi-Newtonian flows, IMA J. Numer. Anal., 33, 4, 1386-1415 (2013) · Zbl 1457.65192
[14] Bustinza, R.; Gatica, G. N., A mixed local discontinuous Galerkin method for a class of nonlinear problems in fluid mechanics, J. Comput. Phys., 207, 2, 427-456 (2005) · Zbl 1213.76118
[15] Cáceres, E.; Gatica, G. N.; Sequeira, F. A., A mixed virtual element method for quasi-Newtonian Stokes flows, SIAM J. Numer. Anal., 56, 1, 317-343 (2018) · Zbl 1380.76033
[16] Gatica, G. N.; Sequeira, F. A., Analysis of an augmented HDG method for a class of quasi-Newtonian Stokes flows, J. Sci. Comput., 65, 3, 1270-1308 (2015) · Zbl 1330.76067
[17] Zheng, X.; Chen, G.; Xie, X., A divergence-free weak Galerkin method for quasi-Newtonian Stokes flows, Sci. China Math., 60, 8, 1515-1528 (2017) · Zbl 1398.76131
[18] Di Pietro, D. A.; Ern, A.; Lemaire, S., An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Meth. Appl. Math., 14, 4, 461-472 (2014) · Zbl 1304.65248
[19] Di Pietro, D. A.; Ern, A., A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Eng., 283, 1-21 (2015) · Zbl 1423.74876
[20] Di Pietro, D. A.; Droniou, J.; Ern, A., A discontinuous-skeletal method for advection-diffusion-reaction on general meshes, SIAM J. Numer. Anal., 53, 5, 2135-2157 (2015) · Zbl 1457.65194
[21] Chave, F.; Di Pietro, D. A.; Marche, F.; Pigeonneau, F., A hybrid high-order method for the Cahn-Hilliard problem in mixed form, SIAM J. Numer. Anal., 54, 3, 1873-1898 (2016) · Zbl 1401.65103
[22] Di Pietro, D. A.; Ern, A.; Linke, A.; Schieweck, F., A discontinuous skeletal method for the viscosity-dependent Stokes problem, Comput. Methods Appl. Mech. Eng., 306, 175-195 (2016) · Zbl 1436.76022
[23] Di Pietro, D. A.; Droniou, J., A hybrid high-order method for Leray-Lions elliptic equations on general meshes, Math. Comput., 86, 307, 2159-2191 (2017) · Zbl 1364.65224
[24] Di Pietro, D. A.; Droniou, J., \(W^{ s,p } \)-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a hybrid high-order discretisation of Leray-Lions problems, Math. Models Methods Appl. Sci., 27, 5, 879-908 (2017) · Zbl 1365.65251
[25] Cockburn, B.; Di Pietro, D. A.; Ern, A., Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM Math. Model Numer. Anal., 50, 3, 635-650 (2016) · Zbl 1341.65045
[26] Boffi, D.; Di Pietro, D. A., Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes, ESAIM Math. Model Numer. Anal., 52, 1, 1-28 (2018) · Zbl 1402.65135
[27] Di Pietro, D. A.; Droniou, J.; Manzini, G., Discontinuous skeletal gradient discretisation methods on polytopal meshes, J. Comput. Phys., 355, 397-425 (2018) · Zbl 1380.65414
[28] Ostwald, W., Ueber die rechnerische darstellung des strukturgebietes der viskosität, Kolloid-Zeitschrift, 47, 2, 176-187 (1929)
[29] Carreau, P. J.; MacDonald, I. F.; Bird, R., A nonlinear viscoelastic model for polymer solutions and melts (ii), Chem. Eng. Sci., 23, 8, 901-911 (1968)
[30] Abraham, F.; Behr, M.; Heinkenschloss, M., Shape optimization in steady blood flow: a numerical study of non-Newtonian effects, Comput. Methods Biomech. Biomed. Eng., 8, 2, 127-137 (2005)
[31] Romero, J. S.; Silva, E. C.N., Non-Newtonian laminar flow machine rotor design by using topology optimization, Struct. Multidiscip. Optim., 55, 5, 1711-1732 (2017)
[32] Di Pietro, D. A.; Ern, A., Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes, IMA J. Numer. Anal., 37, 1, 40-63 (2017) · Zbl 1433.65285
[33] D.A. Di Pietro, R. Tittarelli, Lectures from the Fall 2016 Thematic Quarter at Institut Henri Poincaré, SEMA-SIMAI, Springer.; D.A. Di Pietro, R. Tittarelli, Lectures from the Fall 2016 Thematic Quarter at Institut Henri Poincaré, SEMA-SIMAI, Springer.
[34] Di Pietro, D. A.; Ern, A., Mathematical Aspects of Discontinuous Galerkin Methods. Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], 69 (2012), Springer: Springer Heidelberg · Zbl 1231.65209
[35] Dupont, T.; Scott, R., Polynomial approximation of functions in Sobolev spaces, Math. Comput., 34, 150, 441-463 (1980) · Zbl 0423.65009
[36] Congreve, S.; Houston, P., Two-grid hp-version discontinuous Galerkin finite element methods for quasi-Newtonian fluid flows, Int. J. Numer. Anal. Model., 11, 3, 496-524 (2014) · Zbl 1499.65651
[37] Boffi, D.; Brezzi, F.; Demkowicz, L. F.; Durán, R. G.; Falk, R. S.; Fortin, M., Mixed Finite Elements, Compatibility Conditions, and Applications. Mixed Finite Elements, Compatibility Conditions, and Applications, Lecture Notes in Mathematics, 1939 (2008), Springer-Verlag: Springer-Verlag Berlin
[38] John, V., Finite Element Methods for Incompressible Flow Problems. Finite Element Methods for Incompressible Flow Problems, Springer Series in Computational Mathematics, 51 (2016), Springer: Springer Cham · Zbl 1358.76003
[39] Chen, L., \(i\) FEM: An innovative finite element methods package in MATLAB, preprint (2009), University of California at Irvine: University of California at Irvine Irvine, CA
[40] Talischi, C.; Paulino, G. H.; Pereira, A.; Menezes, I. F.M., Polymesher: a general-purpose mesh generator for polygonal elements written in Matlab, Struct. Multidiscip. Optim., 45, 3, 309-328 (2012) · Zbl 1274.74401
[41] Berrone, S.; Süli, E., Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows, IMA J. Numer. Anal., 28, 2, 382-421 (2008) · Zbl 1136.76030
[42] Heinemann, Z.; Brand, C.; Munka, M.; Chen, Y., Modeling reservoir geometry with irregular grids, SPE Reserv. Eng., 6, 2, 225-232 (1991)
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