On the divergence constraint in mixed finite element methods for incompressible flows. (English) Zbl 1426.76275


76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI


[1] M. Ainsworth and W. Dörfler, Reliable a posteriori error control for nonconformal finite element approximation of Stokes flow, Math. Comp., 74 (2005), pp. 1599–1619. · Zbl 1078.76041
[2] J. Argyris, I. Fried, and D. Scharpf, The TUBA family of plate elements for the matrix displacement method, Aero. J. Roy. Aero. Soc., 72 (1968), pp. 701–709.
[3] D. Arndt, H. Dallmann, and G. Lube, Local projection FEM stabilization for the time-dependent incompressible Navier-Stokes problem, Numer. Methods Partial Differential Equations, 31 (2015), pp. 1224–1250, . · Zbl 1446.76126
[4] D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations, Calcolo, 21 (1984), pp. 337–344, . · Zbl 0593.76039
[5] D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), pp. 1–155, . · Zbl 1185.65204
[6] D. N. Arnold, R. S. Falk, and R. Winther, Finite element exterior calculus: From Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.), 47 (2010), pp. 281–354, . · Zbl 1207.65134
[7] D. N. Arnold and J. Qin, Quadratic velocity/linear pressure Stokes elements, in Advances in Computer Methods for Partial Differential Equations VII, R. Vichnevetsky, D. Knight, and G. Richter, eds., IMACS, 1992, pp. 28–34.
[8] G. Auchmuty and J. C. Alexander, \(L^2\)-well-posedness of 3D div-curl boundary value problems, Quart. Appl. Math., 63 (2005), pp. 479–508, .
[9] T. M. Austin, T. A. Manteuffel, and S. McCormick, A robust multilevel approach for minimizing \(H({div})\)-dominated functionals in an \(H^1\)-conforming finite element space, Numer. Linear Algebra Appl., 11 (2004), pp. 115–140, . · Zbl 1164.65516
[10] G. R. Barrenechea and F. Valentin, Consistent local projection stabilized finite element methods, SIAM J. Numer. Anal., 48 (2010), pp. 1801–1825, . · Zbl 1219.65131
[11] G. R. Barrenechea and F. Valentin, A residual local projection method for the Oseen equation, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1906–1921, . · Zbl 1231.76135
[12] G. R. Barrenechea and F. Valentin, Beyond pressure stabilization: A low-order local projection method for the Oseen equation, Internat. J. Numer. Methods Engrg., 86 (2011), pp. 801–815, . · Zbl 1235.76059
[13] M. Benzi and M. A. Olshanskii, An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput., 28 (2006), pp. 2095–2113, . · Zbl 1126.76028
[14] C. Bernardi and G. Raugel, Analysis of some finite elements for the Stokes problem, Math. Comp., 44 (1985), pp. 71–79, . · Zbl 0563.65075
[15] S. Börm and S. Le Borne, \(H\)-LU factorization in preconditioners for augmented Lagrangian and grad-div stabilized saddle point systems, Internat. J. Numer. Methods Fluids, 68 (2012), pp. 83–98, . · Zbl 1426.76224
[16] C. Brennecke, A. Linke, C. Merdon, and J. Schöberl, Optimal and pressure-independent \(L^2\) velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions, J. Comput. Math., 33 (2015), pp. 191–208, . · Zbl 1340.76024
[17] F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), pp. 217–235, . · Zbl 0599.65072
[18] A. Buffa, C. de Falco, and G. Sangalli, IsoGeometric Analysis: Stable elements for the 2D Stokes equation, Internat. J. Numer. Methods Fluids, 65 (2011), pp. 1407–1422, . · Zbl 1429.76044
[19] A. Buffa, J. Rivas, G. Sangalli, and R. Vázquez, Isogeometric discrete differential forms in three dimensions, SIAM J. Numer. Anal., 49 (2011), pp. 818–844, . · Zbl 1225.65100
[20] Y. V. Bychenkov and E. V. Chizonkov, Optimization of one three-parameter method of solving an algebraic system of the Stokes type, Russian J. Numer. Anal. Math. Modelling, 14 (1999), pp. 429–440, . · Zbl 0947.65038
[21] C. Carstensen and C. Merdon, Computational survey on a posteriori error estimators for the Crouzeix–Raviart nonconforming finite element method for the Stokes problem, Comput. Methods Appl. Math., 14 (2014), pp. 35–54, . · Zbl 1285.65072
[22] M. A. Case, V. J. Ervin, A. Linke, and L. G. Rebholz, A connection between Scott–Vogelius and grad-div stabilized Taylor–Hood FE approximations of the Navier–Stokes equations, SIAM J. Numer. Anal., 49 (2011), pp. 1461–1481, . · Zbl 1244.76021
[23] E. Chizhonkov and M. Olshanskii, On the domain geometry dependence of the LBB condition, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 935–951, . · Zbl 1006.76052
[24] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Stud. Math. Appl. 4, North-Holland, Amsterdam, 1978. · Zbl 0383.65058
[25] B. Cockburn, G. Kanschat, and D. Schötzau, A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J. Sci. Comput., 31 (2007), pp. 61–73, .
[26] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7 (1973), pp. 33–75.
[27] J. de Frutos, B. García-Archilla, V. John, and J. Novo, Grad-div stabilization for the evolutionary Oseen problem with inf-sup stable finite elements, J. Sci. Comput., 66 (2016), pp. 991–1024, . · Zbl 1462.65138
[28] O. Dorok, W. Grambow, and L. Tobiska, Aspects of finite element discretizations for solving the Boussinesq approximation of the Navier-Stokes Equations, in Numerical Fluid Mechanics: Numerical Methods for the Navier-Stokes Equations, Proceedings of the International Workshop held at Heidelberg, 1993, F.-K. Hebeker, R. Rannacher, and G. Wittum, eds., Vieweg+Teubner Verlag, Wiesbaden, 1994, pp. 50–61. · Zbl 0876.76038
[29] J. A. Evans and T. J. R. Hughes, Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations, Math. Models Methods Appl. Sci., 23 (2013), pp. 1421–1478, . · Zbl 1383.76337
[30] R. S. Falk and M. Neilan, Stokes complexes and the construction of stable finite elements with pointwise mass conservation, SIAM J. Numer. Anal., 51 (2013), pp. 1308–1326, . · Zbl 1268.76032
[31] M. Fortin, An analysis of the convergence of mixed finite element methods, RAIRO Anal. Numér., 11 (1977), pp. 341–354. · Zbl 0373.65055
[32] L. P. Franca and T. J. R. Hughes, Two classes of mixed finite element methods, Comput. Methods Appl. Mech. Engrg., 69 (1988), pp. 89–129, . · Zbl 0629.73053
[33] J. Fuhrmann, A. Linke, and H. Langmach, A numerical method for mass conservative coupling between fluid flow and solute transport, Appl. Numer. Math., 61 (2011), pp. 530–553, . · Zbl 1366.76051
[34] J. Fuhrmann, A. Linke, C. Merdon, F. Neumann, T. Streckenbach, H. Baltruschat, and M. Khodayari, Inverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefficients from experimental data, Electrochimica Acta, 211 (2016), pp. 1–10.
[35] K. J. Galvin, A. Linke, L. G. Rebholz, and N. E. Wilson, Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection, Comput. Methods Appl. Mech. Engrg., 237/240 (2012), pp. 166–176, . · Zbl 1253.76057
[36] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer Ser. Comput. Math. 5, Springer-Verlag, Berlin, 1986, . · Zbl 0585.65077
[37] R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Stud. Appl. Math. 9, SIAM, Philadelphia, 1989, . · Zbl 0698.73001
[38] J. Guzmán and M. Neilan, A family of nonconforming elements for the Brinkman problem, IMA J. Numer. Anal., 32 (2012), pp. 1484–1508, . · Zbl 1332.76057
[39] J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements on general triangular meshes, Math. Comp., 83 (2014), pp. 15–36, . · Zbl 1322.76041
[40] T. Heister and G. Rapin, Efficient augmented Lagrangian-type preconditioning for the Oseen problem using grad-div stabilization, Internat. J. Numer. Methods Fluids, 71 (2013), pp. 118–134, .
[41] P. Hood and C. Taylor, Navier–Stokes equations using mixed interpolation, in Finite Element Methods in Flow Problems, J. T. Oden, R. H. Gallagher, O. C. Zienkiewicz, and C. Taylor, eds., University of Alabama in Huntsville Press, 1974, pp. 121–132.
[42] T. J. R. Hughes, L. P. Franca, and M. Balestra, A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput. Methods Appl. Mech. Engrg., 59 (1986), pp. 85–99, . · Zbl 0622.76077
[43] E. W. Jenkins, V. John, A. Linke, and L. G. Rebholz, On the parameter choice in grad-div stabilization for the Stokes equations, Adv. Comput. Math., 40 (2014), pp. 491–516, . · Zbl 1426.76272
[44] V. John and A. Kindl, Numerical studies of finite element variational multiscale methods for turbulent flow simulations, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 841–852, . · Zbl 1406.76029
[45] G. Kanschat and N. Sharma, Divergence-conforming discontinuous Galerkin methods and \(C^0\) interior penalty methods, SIAM J. Numer. Anal., 52 (2014), pp. 1822–1842, . · Zbl 1298.76117
[46] M.-J. Lai and L. L. Schumaker, Spline Functions on Triangulations, Encyclopedia Math. Appl. 110, Cambridge University Press, Cambridge, 2007, .
[47] W. Layton, Introduction to the numerical analysis of incompressible viscous flows, Comput. Sci. Engrg. 6, SIAM, Philadelphia, 2008, . · Zbl 1153.76002
[48] W. Layton, C. C. Manica, M. Neda, M. Olshanskii, and L. G. Rebholz, On the accuracy of the rotation form in simulations of the Navier-Stokes equations, J. Comput. Phys., 228 (2009), pp. 3433–3447, . · Zbl 1161.76030
[49] P. L. Lederer, A. Linke, C. Merdon, and S. Schöberl, Divergence-free reconstructon operators for pressure-robust Stokes discretizations with continuous pressure finite elements, SIAM J. Numer. Anal., 55 (2017), pp. 1291–1314, . · Zbl 1457.65202
[50] A. Linke, A divergence-free velocity reconstruction for incompressible flows, C. R. Math. Acad. Sci. Paris, 350 (2012), pp. 837–840, . · Zbl 1303.76106
[51] A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comput. Methods Appl. Mech. Engrg., 268 (2014), pp. 782–800, . · Zbl 1295.76007
[52] A. Linke, G. Matthies, and L. Tobiska, Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors, ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 289–309, . · Zbl 1381.76186
[53] A. Linke and C. Merdon, On velocity errors due to irrotational forces in the Navier–Stokes momentum balance, J. Comput. Phys., 313 (2016), pp. 654–661, . · Zbl 1349.65627
[54] A. Linke and C. Merdon, Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., 311 (2016), pp. 304–326.
[55] A. Linke, C. Merdon, and W. Wollner, Optimal \({L}^2\) velocity error estimate for a modified pressure-robust Crouzeix–Raviart Stokes element, IMA J. Numer. Anal., 37 (2017), pp. 354–374, .
[56] A. Linke, M. Neilan, L. Rebholz, and N. Wilson, A connection between coupled and penalty protection timestepping schemes with FE spatial discretization for the Navier–Stokes equations, J. Numer. Math., to appear. · Zbl 1453.65331
[57] C. C. Manica, M. Neda, M. Olshanskii, and L. G. Rebholz, Enabling numerical accuracy of Navier-Stokes-\(α\) through deconvolution and enhanced stability, ESAIM Math. Model. Numer. Anal., 45 (2011), pp. 277–307, . · Zbl 1267.76021
[58] K. A. Mardal, X.-C. Tai, and R. Winther, A robust finite element method for Darcy–Stokes flow, SIAM J. Numer. Anal., 40 (2002), pp. 1605–1631, . · Zbl 1037.65120
[59] P. Monk, Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003, . · Zbl 1024.78009
[60] J. Morgan and R. Scott, A nodal basis for \(C^{1}\) piecewise polynomials of degree \(n≥ 5\), Math. Comp., 29 (1975), pp. 736–740. · Zbl 0307.65074
[61] J.-C. Nédélec, Mixed finite elements in \({R}^{3}\), Numer. Math., 35 (1980), pp. 315–341, .
[62] M. Neilan, Discrete and conforming smooth de Rham complexes in three dimensions, Math. Comp., 84 (2015), pp. 2059–2081, . · Zbl 1319.65115
[63] M. Neilan and D. Sap, Stokes elements on cubic meshes yielding divergence-free approximations, Calcolo, 53 (2015), pp. 263–283, . · Zbl 1388.76148
[64] M. A. Olshanskii, A low order Galerkin finite element method for the Navier-Stokes equations of steady incompressible flow: A stabilization issue and iterative methods, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 5515–5536, . · Zbl 1083.76553
[65] M. A. Olshanskii, G. Lube, T. Heister, and J. Löwe, Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 3975–3988, . · Zbl 1231.76161
[66] M. A. Olshanskii and A. Reusken, Grad-div stabilization for Stokes equations, Math. Comp., 73 (2004), pp. 1699–1718, . · Zbl 1051.65103
[67] J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1986. · Zbl 0429.76001
[68] J. Qin, On the Convergence of Some Low Order Mixed Finite Elements for Incompressible Fluids, Ph.D. thesis, Department of Mathematics, Pennsylvania State University, 1994.
[69] P.-A. Raviart and J. M. Thomas, A mixed finite element method for \(2\)nd order elliptic problems, in Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Lecture Notes in Math. 606, Springer, Berlin, 1977, pp. 292–315.
[70] H.-G. Roos, M. Stynes, and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems, 2nd ed., Springer Ser. Comput. Math. 24, Springer-Verlag, Berlin, 2008. · Zbl 1155.65087
[71] L. R. Scott and M. Vogelius, Conforming finite element methods for incompressible and nearly incompressible continua, in Large-Scale Computations in Fluid Mechanics, Part 2 (La Jolla, CA, 1983), Lectures in Appl. Math. 22, AMS, Providence, RI, 1985, pp. 221–244.
[72] L. R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO Modél. Math. Anal. Numér., 19 (1985), pp. 111–143. · Zbl 0608.65013
[73] X.-C. Tai and R. Winther, A discrete de Rham complex with enhanced smoothness, Calcolo, 43 (2006), pp. 287–306, . · Zbl 1168.76311
[74] L. Tobiska and R. Verfürth, Analysis of a streamline diffusion finite element method for the Stokes and Navier–Stokes equations, SIAM J. Numer. Anal., 33 (1996), pp. 107–127, . · Zbl 0843.76052
[75] M. Vogelius, A right-inverse for the divergence operator in spaces of piecewise polynomials. Application to the \(p\)-version of the finite element method, Numer. Math., 41 (1983), pp. 19–37, . · Zbl 0504.65060
[76] J. Wang, Y. Wang, and X. Ye, A robust numerical method for Stokes equations based on divergence-free \(H{\rm(div)}\) finite element methods, SIAM J. Sci. Comput., 31 (2009), pp. 2784–2802, . · Zbl 1407.76074
[77] J. Wang and X. Ye, New finite element methods in computational fluid dynamics by \(H(div)\) elements, SIAM J. Numer. Anal., 45 (2007), pp. 1269–1286, . · Zbl 1138.76049
[78] M. Wohlmuth and M. Dobrowolski, Numerical analysis of Stokes equations with improved LBB dependency, Electron. Trans. Numer. Anal., 32 (2008), pp. 173–189. · Zbl 1171.65082
[79] X. Xie, J. Xu, and G. Xue, Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models, J. Comput. Math., 26 (2008), pp. 437–455. · Zbl 1174.76013
[80] S. Zhang, A new family of stable mixed finite elements for the \(3\)D Stokes equations, Math. Comp., 74 (2005), pp. 543–554, . · Zbl 1085.76042
[81] S. Zhang, A family of \(Q_{k+1,k}× Q_{k,k+1}\) divergence-free finite elements on rectangular grids, SIAM J. Numer. Anal., 47 (2009), pp. 2090–2107, . · Zbl 1406.76055
[82] S. Zhang, Quadratic divergence-free finite elements on Powell-Sabin tetrahedral grids, Calcolo, 48 (2011), pp. 211–244, . · Zbl 1232.65151
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.