Divergence-conforming velocity and vorticity approximations for incompressible fluids obtained with minimal facet coupling. (English) Zbl 07698951
Summary: We introduce two new lowest order methods, a mixed method, and a hybrid discontinuous Galerkin method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi-Douglas-Marini space for approximating the velocity and the lowest order Raviart-Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.
MSC:
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
| 76Dxx | Incompressible viscous fluids |
| 76Mxx | Basic methods in fluid mechanics |
Keywords:
incompressible Stokes equations; mixed finite elements; pressure-robustness; hybrid discontinuous Galerkin methods; discrete Korn inequalityReferences:
| [1] | Arnold, DN; Brezzi, F.; Douglas, J. Jr, PEERS: a new mixed finite element for plane elasticity, Jpn J. Appl. Math., 1.2, 347-367 (1984) · Zbl 0633.73074 · doi:10.1007/BF03167064 |
| [2] | Arnold, DN; Falk, RS; Winther, R., Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comput., 76.260, 1699-1723 (2007) · Zbl 1118.74046 · doi:10.1090/S0025-5718-07-01998-9 |
| [3] | Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications (2013), New York: Springer, New York · Zbl 1277.65092 · doi:10.1007/978-3-642-36519-5 |
| [4] | Boffi, D.; Brezzi, F.; Fortin, M., Reduced symmetry elements in linear elasticity, Commun. Pure Appl. Anal., 8.1, 95-121 (2009) · Zbl 1154.74041 · doi:10.3934/cpaa.2009.8.95 |
| [5] | Brenner, SC, Korn’s inequalities for piecewise H1 vector fields, Math. Comput., 73.247, 1067-1087 (2004) · Zbl 1055.65118 |
| [6] | Cockburn, B.; Gopalakrishnan, J.; Guzmán, J., A new elasticity element made for enforcing weak stress symmetry, Math. Comput., 79.271, 1331-1349 (2010) · Zbl 1369.74078 · doi:10.1090/S0025-5718-10-02343-4 |
| [7] | 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 · doi:10.1137/070706616 |
| [8] | Cockburn, B.; Kanschat, G.; Schötzau, D., A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comput., 74.251, 1067-1095 (2005) · Zbl 1069.76029 |
| [9] | 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 · doi:10.1007/s10915-006-9107-7 |
| [10] | Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. In: RAIRO (Revue Française d’Atomatique, Informatique et Recherche Opérationnelle), Analyse Numérique R-3 (7e année) (1973), pp. 33-76 · Zbl 0302.65087 |
| [11] | Falk, RS, Nonconforming finite element methods for the equations of linear elasticity, Math. Comput., 57.196, 529-550 (1991) · Zbl 0747.73044 · doi:10.1090/S0025-5718-1991-1094947-6 |
| [12] | Farhloul, M.; Fortin, M., A new mixed finite element for the Stokes and elasticity problems, SIAM J. Numer. Anal., 30.4, 971-990 (1993) · Zbl 0777.76051 · doi:10.1137/0730051 |
| [13] | Farhloul, M., Fortin, M.: Review and complements on mixed-hybrid finite element methods for fluid flows. In: Proceedings of the 9th International Congress on Computational and Applied Mathematics (Leuven, 2000), vol. 14.01-2, pp. 301-313 (2002) · Zbl 1134.76383 |
| [14] | Farhloul, M., Mixed and nonconforming finite element methods for the Stokes problem, Can. Appl. Math. Q., 3.4, 399-418 (1995) · Zbl 0846.35100 |
| [15] | Farhloul, M., Fortin, M.: Dual hybrid methods for the elasticity and the Stokes problems: a unified approach. In: Numerische Mathematik, vol. 76, no. 4 , pp. 419-440 (1997) · Zbl 0880.73064 |
| [16] | Fu, G., Uniform auxiliary space preconditioning for hdg methods for elliptic operators with a parameter dependent low order term, SIAM J. Sci. Comput., 43.6, A3912-A3937 (2021) · Zbl 1478.65123 · doi:10.1137/20M1382325 |
| [17] | Fu, G.; Jin, Y.; Qiu, W., Parameter-free superconvergent Hpdivqconforming HDG methods for the Brinkman equations, IMA J. Numer. Anal., 39.2, 957-982 (2019) · Zbl 1466.65187 · doi:10.1093/imanum/dry001 |
| [18] | Girault, V.; Raviart, P-A, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms (2012), New York: Springer, New York |
| [19] | Gopalakrishnan, J.; Guzmán, J., A second elasticity element using the matrix bubble, IMA J. Numer. Anal., 32, 352-372 (2012) · Zbl 1232.74101 · doi:10.1093/imanum/drq047 |
| [20] | Gopalakrishnan, J.; Lederer, PL; Schöberl, J., A mass conserving mixed stress formulation for stokes flow with weakly imposed stress symmetry, SIAM J. Numer. Anal., 58.1, 706-732 (2020) · Zbl 1439.35368 · doi:10.1137/19M1248960 |
| [21] | Gopalakrishnan, J.; Lederer, PL; Schöberl, J., A mass conserving mixed stress formulation for the Stokes equations, IMA J. Numer. Anal., 40.3, 1838-1874 (2019) · Zbl 1466.65189 |
| [22] | Kogler, L., Lederer, P.L., Schöberl, J.: A conforming auxiliary space preconditioner for the mass conserving stress-yielding method. (2022). arXiv: 2207.08481 [math.NA] |
| [23] | Landau, L.D., Lifshitz, E.M.: Fluid mechanics. Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, vol. 6. Pergamon Press, London (1959) · JFM 64.0887.01 |
| [24] | Lederer, P.L.: A Mass Conserving Mixed Stress Formulation for Incompressible Flows. PhD thesis. Technical University of Vienna (2019) |
| [25] | Lederer, PL; Lehrenfeld, C.; Schöberl, J., Hybrid discontinuous Galerkin methods with relaxed Hpdivq-conformity for incompressible flows. Part II. ESAIM, Math. Model. Numer. Anal., 53.2, 503-522 (2019) · Zbl 1434.35058 · doi:10.1051/m2an/2018054 |
| [26] | Lederer, PL; Lehrenfeld, C.; Schöberl, J., Hybrid discontinuous Galerkin methods with relaxed Hpdivq-conformity for incompressible flows. Part I, SIAM J. Numer. Anal., 56.4, 2070-2094 (2018) · Zbl 1402.35209 · doi:10.1137/17M1138078 |
| [27] | Lederer, PL; Linke, A.; Merdon, C.; Schöberl, J., Divergence-free reconstruction operators for pressure-robust stokes discretizations with continuous pressure finite elements, SIAM J. Numer. Anal., 55.3, 1291-1314 (2017) · Zbl 1457.65202 · doi:10.1137/16M1089964 |
| [28] | Lederer, P., Kogler, L., Gopalakrishnan, J., Schöberl., J.: Computational results and python files for the work ”Divergence-conforming velocity and vorticity approximations for incompressible fluids obtained with minimal facet coupling (2023). doi:10.5281/zenodo.7767775 |
| [29] | Lehrenfeld, C.; Schöberl, J., High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows, Comput. Methods Appl. Mech. Eng., 307, 339-361 (2016) · Zbl 1439.76081 · doi:10.1016/j.cma.2016.04.025 |
| [30] | Linke, A., On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comput. Methods Appl. Mech. Eng., 268, 782-800 (2014) · Zbl 1295.76007 · doi:10.1016/j.cma.2013.10.011 |
| [31] | Mardal, K-A; Winther, R., An observation on Korn’s inequality for nonconforming finite element methods, Math. Comput., 75.253, 1-6 (2006) · Zbl 1086.65112 |
| [32] | Schöberl, J., NETGEN an advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci., 1.1, 41-52 (1997) · Zbl 0883.68130 |
| [33] | Schöberl, J.: C++11 Implementation of Finite Elements in NGSolve. Technical report ASC-2014-30. Institute for Analysis and Scientific Computing, (2014) |
| [34] | Stenberg, R., A family of mixed finite elements for the elasticity problem, Numer. Math., 53.5, 513-538 (1988) · Zbl 0632.73063 · doi:10.1007/BF01397550 |
| [35] | Tai, X.-C., Winther, R.: A discrete de Rham complex with enhanced smoothness. Calcolo 43.4, 287-306 (2006) · Zbl 1168.76311 |
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