A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations. (English) Zbl 1216.76034

Summary: We present a high-order discontinuous Galerkin discretization of the unsteady incompressible Navier–Stokes equations in convection-dominated flows using triangular and tetrahedral meshes. The scheme is based on a semi-explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the Stokes operator. The nonlinear term is discretized in divergence form by using the local Lax-Friedrichs fluxes; thus, local conservativity is inherent. Spatial discretization of the Stokes operator has employed both equal-order \((P_{k} - P_{k})\) and mixed-order \((P_{k} - P_{k - 1})\) velocity and pressure approximations. A second-order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic Helmholtz equation for each component of the velocity and a consistent Poisson equation for the pressure. The consistent Poisson operator is replaced by an equivalent (in stability and convergence) operator, namely that arising from the interior penalty discretization of the standard Poisson operator with appropriate boundary conditions. This yields a simpler and more efficient method, characterized by a compact stencil size.
We show the temporal and spatial behavior of the method by solving some popular benchmarking tests. For an unsteady Stokes problem, second-order temporal convergence is obtained, while for the Taylor vortex test problem on both semi-structured and fully unstructured triangular meshes, spectral convergence with respect to the polynomial degree k is obtained. By studying the Orr–Sommerfeld stability problem, we demonstrate that the \(P_{k} - P_{k}\) method yields a stable solution, while the \(P_{k} - P_{k - 1}\) formulation leads to unphysical instability. The good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder. We conclude that the proposed discontinuous Galerkin method with the \(P_{k} - P_{k}\) formulation is an efficient scheme for accurate and stable solution of the unsteady Navier–Stokes equations in convection-dominated flows.


76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids


Gmsh; PETSc
Full Text: DOI


[1] Arnold, D., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 742-760, (1982) · Zbl 0482.65060
[2] Arnold, D.N.; Brezzi, F.; Cockburn, B.; Marini, L.D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. numer. anal., 39, 1749-1779, (2002) · Zbl 1008.65080
[3] Castillo, P., Performance of discontinuous Galerkin methods for elliptic pdes, SIAM J. sci. comput., 24, 524-547, (2002) · Zbl 1021.65054
[4] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer-Verlag Heidelberg · Zbl 0658.76001
[5] W. Couzy, Spectral element discretization of the unsteady Navier-Stokes equations and its iterative solution on parallel computers, Thesis No. 1380, École Polytechnique Fédérate de Lausanne, 1995.
[6] Dubiner, M., Spectral methods on triangle and other domains, J. sci. comput., 6, 345-390, (1991) · Zbl 0742.76059
[7] Fischer, P.F., An overlapping Schwarz method for spectral element solution of the incompressible navier – stokes equations, J. comput. phys., 133, 84-101, (1997) · Zbl 0904.76057
[8] C. Geuzaine, J.F. Remacle, Gmsh Reference Manual, Edition 1.12. Available from: <http://www.geuz.org.org/gmsh/>, 2003.
[9] Hesthaven, J.S., From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. numer. anal., 35, 655-676, (1998) · Zbl 0933.41004
[10] Hesthaven, J.S.; Teng, C.H., Stable spectral methods on tetrahedral elements, SIAM J. sci. comput., 21, 2352-2380, (2000) · Zbl 0959.65112
[11] Hesthaven, J.S.; Warburton, T., Nodal high-order methods on unstructured grids. I. time-domain solution of maxwell’s equations, J. comput. phys., 181, 186-221, (2002) · Zbl 1014.78016
[12] Heywood, G.J.; Rannacher, R.; Turek, S., Artificial boundaries and flux and pressure conditions for the incompressible navier – stokes equations, Int. J. numer. methods fluids, 22, 325-352, (1996) · Zbl 0863.76016
[13] Karniadakis, G.E.; Israeli, M.; Orszag, S.A., High-order splitting methods for the incompressible navier – stokes equations, J. comput. phys., 97, 414-443, (1991) · Zbl 0738.76050
[14] Karniadakis, G.E.; Sherwin, S.J., Spectral/hp element methods for computational fluid dynamics, (2005), Oxford Univ. Press London · Zbl 1116.76002
[15] Maday, Y.; Patera, A.T.; Rønquist, E.M., An operator-integration-factor splitting method for time-dependent problems: application to incompressibel fluid flow, J. sci. comput., 5, 263-292, (1990) · Zbl 0724.76070
[16] Warburton, T.; Pavarino, L.F.; Hesthaven, J.S., A pseudo-spectral scheme for the incompressible navier – stokes equations using unstructured nodal elements, J. comput. phys., 164, 1-21, (2000) · Zbl 0961.76063
[17] Koornwinder, T., Theory and application of special functions, (1975), Academic Press New York, pp. 435-495
[18] Schötzau, D.; Schwab, C.; Toselli, A., Mixed hp-DGFEM incompressible flows, SIAM J. numer. anal., 40, 2171-2194, (2003) · Zbl 1055.76032
[19] Schötzau, D.; Schwab, C.; Toselli, A., Mixed hp-DGFEM for incompressible flows. II. geometric edge meshes, IMA J. numer. anal., 24, 273-308, (2004) · Zbl 1114.76046
[20] Hansbo, P.; Larson, M.G., Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by nitsche’s method, Comput. methods appl. mech. eng., 191, 1895-1908, (2002) · Zbl 1098.74693
[21] Cockburn, B.; Kanschat, G.; Schötzau, D.; Schwab, C., Local discontinuous Galerkin methods for the Stokes system, SIAM J. numer. anal., 40, 319-343, (2002) · Zbl 1032.65127
[22] Cockburn, B.; Kanschat, G.; Schötzau, D., The local discontinuous Galerkin method for the Oseen equations, Math. comput., 73, 569-593, (2004) · Zbl 1066.76036
[23] 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
[24] Cockburn, B.; Shu, C., Runge – kutta discontinuous Galerkin methods for convection-dominated flows, J. sci. comput., 16, 173-261, (2001) · Zbl 1065.76135
[25] Cockburn, B.; Shu, C., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. numer. anal., 35, 2440-2463, (1998) · Zbl 0927.65118
[26] Hughes, T.J.R.; Engel, G.; Mazzei, L., A comparison of discontinuous and continuous Galerkin methods based on error estimates, conservation, robustness and efficiency, (), 135-146 · Zbl 0946.65109
[27] Darekar, R.M.; Sherwin, S.J., Flow past a square-section cylinder with a wavy stagnation face, J. fluid mech., 426, 263-295, (2001) · Zbl 1016.76015
[28] Barkley, D.; Henderson, R.D., Three-dimensional Floquet stability analysis of the wake of a circular cylinder, J. fluid mech., 322, 215-241, (1996) · Zbl 0882.76028
[29] Girault, V.; Riviére, B.; Wheeler, M.F., A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and navier – stokes problems, Math. comput., 74, 53-84, (2005) · Zbl 1057.35029
[30] Henriksen, M.O.; Holmen, J., Algebraic splitting for incompressible navier – stokes equations, J. comput. phys., 175, 438-453, (2002) · Zbl 1059.76045
[31] Shahbazi, K., An explicit expression for the penalty parameter of the interior penalty method, J. comput. phys., 205, 401-407, (2005) · Zbl 1072.65149
[32] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible navier – stokes equations, J. comput. phys., 131, 267-279, (1997) · Zbl 0871.76040
[33] S. Beuchler, J. Schöberl, New shape functions for triangular p-FEM using integrated Jacobi polynomials, Technical Report 2004-18, Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria. Available from: <http://www.ricam.oeaw.ac.at/publications/reports/04/rep04-18.pdf>.
[34] Okajima, A., Strouhal numbers of rectangular cylinders, J. fluid mech., 123, 379-398, (1982)
[35] J. Schöberl, J. Melenk, C. Pechstein, S. Zaglmayr, Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements. Available from: <http://www.hpfem.jku.at/index.html?/joachim>, 2005.
[36] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc home page. Available from: http://www.mcs.anl.gov/petsc, 2001.
[37] S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, H. Zhang, PETSc Users Manual, ANL-95/11 - Revision 2.1.5, Argonne National Laboratory, 2004.
[38] Perot, J.B., An analysis of the fractional step method, J. comput. phys., 108, 51-58, (1993) · Zbl 0778.76064
[39] J. Remacle, AOMD home page. Available from: <http://www.scorec.rpi.edu/AOMD/>, 2002.
[40] Solin, P.; Segeth, K.; Dolezel, I., Higher-order finite element methods, Studies in advanced mathematics, (2004), Chapman & Hall/CRC Boca Raton, FL
[41] Szegö, G., Orthogonal polynomials, (1939), Am. Math. Soc. Providence · JFM 65.0278.03
[42] Wilhelm, D.; Kleiser, L., Stability analysis for different formulations of the nonlinear term in PN−PN−2 spectral element discretizations of the navier – stokes equations, J. comput. phys., 174, 306-326, (2001) · Zbl 1106.76414
[43] Chorin, A.J., On the convergence of discrete approximation to the navier – stokes equations, Math. comput., 23, 341-353, (1969) · Zbl 0184.20103
[44] Temam, R., Une mthode d’approximation de la solution des quations de navier – stokes, Bulletin de la socit mathmatique de France, 96, 115-152, (1968) · Zbl 0181.18903
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