## An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations.(English)Zbl 1391.76353

Summary: We present an implicit high-order hybridizable discontinuous Galerkin method for the steady-state and time-dependent incompressible Navier-Stokes equations. The method is devised by using the discontinuous Galerkin discretization for a velocity gradient-pressure-velocity formulation of the incompressible Navier-Stokes equations with a special choice of the numerical traces. The method possesses several unique features which distinguish itself from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Moreover, if the augmented Lagrangian method is used to solve the linearized system, the globally coupled unknowns become the approximate trace of the velocity only. Second, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of $$k + 1$$ in the $$L^{2}$$-norm, when polynomials of degree $$k\geqslant 0$$ are used for all components of the approximate solution. And third, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, $$H(\mathrm{div})$$-conforming, and converges with order $$k + 2$$ for $$k\geqslant 1$$ and with order 1 for $$k = 0$$ in the $$L^{2}$$-norm. Moreover, a novel and systematic way is proposed for imposing boundary conditions for the stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the method. This can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method. Extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35Q30 Navier-Stokes equations
Full Text:

### References:

 [1] Arnold, D.N.; Brezzi, F.; Cockburn, B.; Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. numer. anal., 39, 5, 1749-1779, (2001) · Zbl 1008.65080 [2] Baker, G.A.; Jureidini, W.N.; Karakashian, O.A., Piecewise solenoidal vector fields and the Stokes problem, SIAM J. numer. anal., 27, 1466-1485, (1990) · Zbl 0719.76047 [3] Bassi, F.; Crivellini, A.; Di Pietro, D.A.; Rebay, S., An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible navier – stokes equation, J. comput. phys., 218, 2, 794-815, (2006) · Zbl 1158.76313 [4] Carrero, J.; Cockburn, B.; Schötzau, D., Hybridized globally divergence-free LDG methods. I. the Stokes problem, Math. comput., 75, 533-563, (2006) · Zbl 1087.76061 [5] Cockburn, B., Discontinuous Galerkin methods, ZAMM Z. angew. math. mech., 83, 731-754, (2003) · Zbl 1036.65079 [6] Cockburn, B., Discontinuous Galerkin methods for computational fluid dynamics, (), 91-123 [7] Cockburn, B.; Dong, B.; Guzmán, J., A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. comput., 77, 1887-1916, (2008) · Zbl 1198.65193 [8] Cockburn, B.; Dong, B.; Guzmán, J.; Restelli, M.; Sacco, R., Superconvergent and optimally convergent LDG-hybridizable discontinuous Galerkin methods for convection – diffusion – reaction problems, SIAM J. sci. comput., 31, 3827-3846, (2009) · Zbl 1200.65093 [9] Cockburn, B.; Gopalakrishnan, J., Incompressible finite elements via hybridization. part I: the Stokes system in two space dimensions, SIAM J. numer. anal., 43, 4, 1627-1650, (2005) · Zbl 1145.76402 [10] Cockburn, B.; Gopalakrishnan, J., Incompressible finite elements via hybridization. part II: the Stokes system in three space dimensions, SIAM J. numer. anal., 43, 4, 1651-1672, (2005) · Zbl 1145.76403 [11] Cockburn, B.; Gopalakrishnan, J., The derivation of hybridizable discontinuous Galerkin methods for Stokes flow, SIAM J. numer. anal., 47, 1092-1125, (2009) · Zbl 1279.76016 [12] Cockburn, B.; Gopalakrishnan, J.; Guzmán, J., A new elasticity element made for enforcing weak stress symmetry, Math. comput., 79, 1331-1349, (2010) · Zbl 1369.74078 [13] Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order elliptic problems, SIAM J. numer. anal., 47, 1319-1365, (2009) · Zbl 1205.65312 [14] B. Cockburn, J. Gopalakrishnan, N.C. Nguyen, J. Peraire, F.-J. Sayas, Analysis of an HDG method for Stokes flow, Math. Comput. Posted on: 2nd September, 2010. PII S 0025-5718-2010-02410-X-1, in press. · Zbl 1410.76164 [15] Cockburn, B.; Gopalakrishnan, J.; Sayas, F.-J., A projection-based error analysis of HDG methods, Math. comput., 79, 1351-1367, (2010) · Zbl 1197.65173 [16] Cockburn, B.; Guzmán, J.; Soon, S.-C.; Stolarski, H.K., An analysis of the embedded discontinuous Galerkin method for second-order elliptic problems, SIAM J. numer. anal., 47, 2686-2707, (2009) · Zbl 1211.65153 [17] Cockburn, B.; Guzmán, J.; Wang, H., Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. comput., 78, 1-24, (2009) · Zbl 1198.65194 [18] Cockburn, B.; Kanschat, G.; Schötzau, D., A locally conservative LDG method for the incompressible navier – stokes equations, Math. comput., 74, 1067-1095, (2005) · Zbl 1069.76029 [19] Cockburn, B.; Kanschat, G.; Schötzau, D., A note on discontinuous Galerkin divergence-free solutions of the navier – stokes equations, J. sci. comput., 31, 61-73, (2007) · Zbl 1151.76527 [20] Cockburn, B.; Kanschat, G.; Schötzau, D.; Schwab, C., Local discontinuous Galerkin methods for the Stokes system, SIAM J. numer. anal., 40, 1, 319-343, (2002) · Zbl 1032.65127 [21] Cockburn, B.; Shu, C.-W., Runge – kutta discontinuous Galerkin methods for convection-dominated problems, J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135 [22] Denham, M.K.; Patrick, M.A., Laminar flow over a downstream-facing step in a two-dimensional flow channel, Trans. inst. chem. eng., 52, 361-367, (1974) [23] Fortin, M.; Glowinski, R., Augmented Lagrangian methods, Studies in mathematics and its applications, vol. 15, (1983), North-Holland Publishing Co. Amsterdam, (Applications to the numerical solution of boundary value problems. Translated from the French by B. Hunt and D.C. Spicer) [24] Gelfgat, A. Yu.; Bar-Yoseph, P.Z.; Yarin, A.L., Stability of multiple steady states of convection in laterally heated cavities, J. fluid mech., 388, 315-334, (1999) · Zbl 0958.76022 [25] Ghia, U.; Ghia, K.N.; Shin, C.T., High-re solutions for incompressible flow using the navier – stokes equations and a multigrid method, J. comput. phys., 48, 387-411, (1982) · Zbl 0511.76031 [26] Güzey, S.; Cockburn, B.; Stolarski, H., The embedded discontinuous Galerkin methods: application to linear shells problems, Int. J. numer. meth. eng., 70, 757-790, (2007) · Zbl 1194.74403 [27] Hughes, T.J.R.; Scovazzi, G.; Bochev, P.; Buffa, A., A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method, Comput. meth. appl. mech. eng., 195, 2761-2787, (2006) · Zbl 1124.76027 [28] Karakashian, O.A.; Jureidini, W.N., A nonconforming finite element method for the stationary navier – stokes equations, SIAM J. numer. anal., 35, 93-120, (1998) · Zbl 0933.76047 [29] Karakashian, O.A.; Katsaounis, T., A discontinuous Galerkin method for the incompressible navier – stokes equations, (), 157-166 · Zbl 0991.76044 [30] Kovasznay, L.I.G., Laminar flow behind two-dimensional grid, Proc. Cambridge philos. soc., 44, 58-62, (1948) · Zbl 0030.22902 [31] Labeur, R.J.; Wells, G.N., A Galerkin interface stabilisation method for the advection – diffusion and incompressible navier – stokes equations, Comput. meth. appl. mech. eng., 196, 4985-5000, (2007) · Zbl 1173.76344 [32] Liu, J.-G.; Shu, C.-W., A high-order discontinuous Galerkin method for 2D incompressible flows, J. comput. phys., 160, 577-596, (2000) · Zbl 0963.76069 [33] Montlaur, A.; Fernández-Méndez, S.; Huerta, A., Discontinuous Galerkin methods for the Stokes equations using divergence-free approximations internat, J. numer. meth. fluids, 57, 1071-1092, (2008) · Zbl 1338.76062 [34] Montlaur, A.; Fernández-Méndez, S.; Peraire, J.; Huerta, A.; approximations, Discontinuous Galerkin methods for the Navier-Stokes equations using solenoidal, I, NT. J. numer. meth. fluids, 64, 549-564, (2010) [35] Nédélec, J.-C., Mixed finite elements in R^{3}, Numer. math., 35, 315-341, (1980) · Zbl 0419.65069 [36] Nédélec, J.-C., A new family of mixed finite elements in R3, Numer. math., 50, 57-81, (1986) · Zbl 0625.65107 [37] Nguyen, N.C.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for linear convection – diffusion equations, J. comput. phys., 228, 3232-3254, (2009) · Zbl 1187.65110 [38] Nguyen, N.C.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection – diffusion equations, J. comput. phys., 228, 8841-8855, (2009) · Zbl 1177.65150 [39] Nguyen, N.C.; Peraire, J.; Cockburn, B., A hybridizable discontinuous Galerkin method for Stokes flow, Comput. meth. appl. mech. eng., 199, 582-597, (2010) · Zbl 1227.76036 [40] Nguyen, N.C.; Peraire, J.; Cockburn, B., A comparison of HDG methods for Stokes flow, J. sci. comput., 45, 215-237, (2010) · Zbl 1203.76079 [41] N.C. Nguyen, J. Peraire, B. Cockburn, Hybridizable discontinuous Galerkin methods, in: Proceedings of the International Conference on Spectral and High Order Methods, Trondheim, Norway, June 2009, Lecture Notes in Computational Science and Engineering, 76 (2011) 63-84. · Zbl 1216.65160 [42] Patera, A.T., A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. comput. phys., 54, 468-488, (1984) · Zbl 0535.76035 [43] J. Peraire, N.C. Nguyen, B. Cockburn, A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations (AIAA Paper 2010-363), in: Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, Florida, January 2010. [44] Shahbazi, K.; F Fischer, P.; Ethier, C.R., A high-order discontinuous Galerkin method for the unsteady incompressible navier – stokes equations, J. comput. phys., 222, 391-407, (2007) · Zbl 1216.76034 [45] Soon, S.-C.; Cockburn, B.; Stolarski, H.K., A hybridizable discontinuous Galerkin method for linear elasticity internat, J. numer. meth. eng., 80, 1058-1092, (2009) · Zbl 1176.74196 [46] M.P. Ueckermann, P.F.J. Lermusiaux, High order schemes for 2D unsteady biogeochemical ocean models, Ocean Dynam. in press, doi:10.1007/s10236-010-0351-x. · Zbl 1351.76083 [47] Wang, J.; Ye, X., New finite element methods in computational fluid dynamics by H(div) elements, SIAM J. numer. anal., 45, 1269-1286, (2007) · Zbl 1138.76049 [48] Wang, J.; Wang, X.; Ye, X., Finite element methods for the navier – stokes equations by H(div) elements, J. comput. math., 26, 410-436, (2008) · Zbl 1174.76012 [49] Wang, J.; Wang, X.; Ye, X., A robust numerical method for Stokes equations based on divergence-free H(div) finite element methods, SIAM J. sci. comput., 31, 2784-2802, (2009) · Zbl 1407.76074
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.