## High order exactly divergence-free Hybrid Discontinuous Galerkin methods for unsteady incompressible flows.(English)Zbl 1439.76081

Summary: In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier-Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency of the discretization method is the distinction between stiff linear parts and less stiff non-linear parts with respect to their temporal and spatial treatment.
Exploiting the flexibility of operator-splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub-problems: a corresponding hyperbolic transport problem and an unsteady Stokes problem.
For the hyperbolic transport problem a spatial discretization with an Upwind Discontinuous Galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation. The treatment of the Stokes part involves the solution of linear systems. In this case a discretization with Hybrid Discontinuous Galerkin methods is better suited. We consider such a discretization for the Stokes part with two important features: $$H(\text{div})$$-conforming finite elements to guarantee exactly divergence-free velocity solutions and a projection operator which reduces the number of globally coupled unknowns. We present the method, discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems.

### MSC:

 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

### Software:

NGSolve; FEATFLOW; ANSYS-CFX; OpenFOAM; PARDISO
Full Text:

### References:

 [1] Reed, W. H.; Hill, T., (Triangular Mesh Methods for the Neutron Transport Equation, Los Alamos Report LA-UR-73-479 (1973)) [2] Lesaint, P.; Raviart, P. A., On a finite element method for solving the neutron transport equation, Math. Aspects Finite Elem. Partial Differ. Equ., 33, 89-123 (1974) [3] Johnson, C.; Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46, 173, 1-26 (1986) · Zbl 0618.65105 [4] Bassi, F.; Rebay, S., High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys., 138, 2, 251-285 (1997) · Zbl 0902.76056 [5] Bassi, F.; Rebay, S.; Mariotti, G.; Pedinotti, S.; Savini, M., A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, (Proceedings of 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics (1997), Technologisch Instituut: Technologisch Instituut Antwerpen, Belgium), 99-108 [6] Cockburn, B.; Shu, C. W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 2, 199-224 (1998) · Zbl 0920.65059 [7] 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 (2002) · Zbl 1008.65080 [8] Houston, P.; Schwab, C.; Süli, E., Discontinuous hp-finite element methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 39, 6, 2133-2163 (2002) · Zbl 1015.65067 [9] Rivière, B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation (2008), Society for Industrial and Applied Mathematics · Zbl 1153.65112 [10] Di Pietro, D. A.; Ern, A., Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 (2012), Springer Science & Business Media · Zbl 1231.65209 [11] Hesthaven, J. S.; Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (2007), Springer Science & Business Media [12] Karniadakis, G.; Sherwin, S., Spectral/hp Element Methods for Computational Fluid Dynamics (2013), Oxford University Press · Zbl 1256.76003 [13] Girault, V.; Raviart, P. A., Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, vol. 5 (2012), Springer Science & Business Media [14] Elman, H. C.; Silvester, D. J.; Wathen, A. J., Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics (2014), Oxford University Press · Zbl 1304.76002 [15] Donea, J.; Huerta, A., Finite Element Methods for Flow Problems (2003), John Wiley & Sons [16] Toselli, A., hp discontinuous Galerkin approximations for the Stokes problem, Math. Models Methods Appl. Sci., 12, 11, 565-1597 (2002) · Zbl 1041.76045 [17] Schötzau, D.; Schwab, C.; Toselli, A., Mixed hp-DGFEM for incompressible flows, SIAM J. Numer. Anal., 40, 6, 2171-2194 (2002) · Zbl 1055.76032 [18] Girault, V.; Rivière, B.; Wheeler, M., A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems, Math. Comp., 74, 249, 53-84 (2005) · Zbl 1057.35029 [19] Cockburn, B.; Kanschat, G.; Schötzau, D., A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comp., 74, 251, 1067-1095 (2005) · Zbl 1069.76029 [20] Cockburn, B.; Kanschat, G.; Schötzau, D., A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J. Sci. Comput., 31, 1-2, 61-73 (2007) · Zbl 1151.76527 [21] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods, vol. 15 (2012), Springer Science & Business Media · Zbl 1009.65067 [22] Lehrenfeld, C., Hybrid discontinuous Galerkin methods for solving incompressible flow problems, Rheinisch-Westfalischen Technischen Hochschule Aachen (2010) [23] Egger, H.; Schöberl, J., A hybrid mixed discontinuous Galerkin method for convection-diffusion problems, J. Numer. Anal (2008) [24] Nguyen, N. C.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys., 228, 9, 3232-3254 (2009) · Zbl 1187.65110 [25] 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, 2, 1319-1365 (2009) · Zbl 1205.65312 [26] Cockburn, B.; Gopalakrishnan, J.; Nguyen, N.; Peraire, J.; Sayas, F. J., Analysis of HDG methods for Stokes flow, Math. Comp., 80, 274, 723-760 (2011) · Zbl 1410.76164 [27] Cesmelioglu, A.; Cockburn, B.; Nguyen, N. C.; Peraire, J., Analysis of HDG methods for Oseen equations, J. Sci. Comput., 55, 2, 392-431 (2013) · Zbl 1366.76048 [28] Zhai, Q.; Zhang, R.; Wang, X., A hybridized weak Galerkin finite element scheme for the Stokes equations, Sci. China Math., 1-18 (2015) [29] Pietro, D. A.D.; Ern, A.; Lemaire, S., An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Methods Appl. Math., 14, 4, 461-472 (2014) · Zbl 1304.65248 [30] Di Pietro, D.; Ern, A.; Lemaire, S., A Review of Hybrid High-Order Methods: Formulations, Cmputational Aspects, Comparison with Other Methods, Technical Report HAL-01163569 (2015), September [31] Cockburn, B.; Pietro, D. D.; Ern, A., Bridging the Hybrid High-Order and Hybridizable Discontinuous Galerkin Methods, Technical Report HAL-01115318 (July,2015) [32] Egger, H.; Waluga, C., hp analysis of a hybrid DG method for Stokes flow, IMA J. Numer. Anal., drs018 (2012) [33] Nguyen, N.; Peraire, J.; Cockburn, B., A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Engrg., 199, 9, 582-597 (2010) · Zbl 1227.76036 [34] Carrero, J.; Cockburn, B.; Schötzau, D., Hybridized globally divergence-free LDG methods. Part i: The Stokes problem, Math. Comp., 75, 254, 533-563 (2006) · Zbl 1087.76061 [35] 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 [36] 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 [37] Könnö, J.; Stenberg, R., Numerical computations with H(div)-finite elements for the Brinkman problem, Comput. Geosci., 16, 1, 139-158 (2012) · Zbl 1348.76100 [39] Oikawa, I., A hybridized discontinuous Galerkin method with reduced stabilization, J. Sci. Comput., 65, 1, 327-340 (2015) · Zbl 1331.65162 [41] Qiu, W.; Shi, K., A superconvergent HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes, CoRR (2015) [42] Kirby, R. M.; Sherwin, S. J.; Cockburn, B., To CG or to HDG: a comparative study, J. Sci. Comput., 51, 1, 183-212 (2012) · Zbl 1244.65174 [43] Huerta, A.; Angeloski, A.; Roca, X.; Peraire, J., Efficiency of high-order elements for continuous and discontinuous Galerkin methods, Internat. J. Numer. Methods Engrg., 96, 9, 529-560 (2013) · Zbl 1352.65512 [44] Ascher, U. M.; Ruuth, S. J.; Wetton, B. T., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32, 3, 797-823 (1995) · Zbl 0841.65081 [45] Ascher, U. M.; Ruuth, S. J.; Spiteri, R. J., Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25, 2, 151-167 (1997) · Zbl 0896.65061 [46] Kanevsky, A.; Carpenter, M. H.; Gottlieb, D.; Hesthaven, J. S., Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes, J. Comput. Phys., 225, 2, 1753-1781 (2007) · Zbl 1123.65097 [47] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 3, 506-517 (1968) · Zbl 0184.38503 [48] Maday, Y.; Patera, A. T.; Rønquist, E. M., An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow, J. Sci. Comput., 5, 4, 263-292 (1990) · Zbl 0724.76070 [49] Chrispell, J.; Ervin, V.; Jenkins, E., A fractional step $$\theta$$-method for convection-diffusion problems, J. Math. Anal. Appl., 333, 1, 204-218 (2007) · Zbl 1156.65074 [50] Zaglmayr, S., High order finite element methods for electromagnetic field computation (2006), JKU Linz, (Ph.D. thesis) [51] Arnold, D. N., An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19, 4, 742-760 (1982) · Zbl 0482.65060 [52] Schöberl, J.; Lehrenfeld, C., Domain decomposition preconditioning for high order hybrid discontinuous Galerkin methods on tetrahedral meshes, (Advanced Finite Element Methods and Applications (2013), Springer), 27-56 · Zbl 1263.65120 [53] Brezzi, F.; Manzini, G.; Marini, D.; Pietra, P.; Russo, A., Discontinuous finite elements for diffusion problems, (Atti Convegno in onore di F. Brioschi (Milano 1997) (1999), Istituto Lombardo, Accademia di Scienze e Lettere), 197-217 [55] Linke, A., On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comput. Methods Appl. Mech. Engrg., 268, 782-800 (2014) · Zbl 1295.76007 [56] Dubiner, M., Spectral methods on triangles and other domains, J. Sci. Comput., 6, 4, 345-390 (1991) · Zbl 0742.76059 [57] Glowinski, R., Finite element methods for incompressible viscous flow, Handb. Numer. Anal., 9, 3-1176 (2003) · Zbl 1040.76001 [58] Chorin, A. J., The numerical solution of the Navier-Stokes equations for an incompressible fluid, Bull. Amer. Math. Soc., 73, 6, 928-931 (1967) · Zbl 0168.46501 [59] Schäfer, M.; Turek, S.; Durst, F.; Krause, E.; Rannacher, R., Benchmark computations of laminar flow around a cylinder, (Flow Simulation with High-performance Computers II (1996)), 547-566 · Zbl 0874.76070 [60] Bayraktar, E.; Mierka, O.; Turek, S., Benchmark computations of 3D laminar flow around a cylinder with CFX, OpenFOAM and FeatFlow, Int. J. Comput. Sci. Eng., 7, 3, 253-266 (2012) [61] John, V., On the efficiency of linearization schemes and coupled multigrid methods in the simulation of a 3d flow around a cylinder, Internat. J. Numer. Methods Fluids, 50, 7, 845-862 (2006) · Zbl 1086.76039 [63] Schöberl, J., C++11 Implementation of Finite Elements in NGSolve, Technical Report ASC-2014-30 (2014), Institute for Analysis and Scientific Computing, September, http://www.asc.tuwien.ac.at/ schoeberl/wiki/publications/ngs-cpp11.pdf [64] Kovasznay, L., Laminar flow behind a two-dimensional grid, (Mathematical Proceedings of the Cambridge Philosophical Society, vol. 44 (1948), Cambridge Univ Press), 58-62 · Zbl 0030.22902 [66] Schenk, O.; Gärtner, K., Solving unsymmetric sparse systems of linear equations with PARDISO, J. Future Gener. Comput. Syst., 20, 475-487 (2004)
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.