## Hybrid discontinuous Galerkin methods with relaxed $$H(\operatorname{div})$$-conformity for incompressible flows. I.(English)Zbl 1402.35209

The authors deal with the Stokes problem $\begin{cases} -\nu \Delta u + \nabla p = f \;\text{ in } \Omega, &\\ \operatorname{div} u=0 \;\text{ in } \Omega,&\\ u=0 \;\text{ on } \Gamma_D,& \end{cases}$ where $$\Gamma_D$$ is a subset of $$\partial \Omega$$. They propose a new discretization method which is an improved version of the recent (2016) method reported by two of the authors of the present paper. This new method is based on an $$H(\operatorname{div})$$-conforming finite element space and a hybrid discontinuous Galerkin formulation of the viscous forces. The authors present this method in detail, explain why it is more advantageous, perform a thorough $$h$$-version error analysis, and provide some numerical examples.

### MSC:

 35Q30 Navier-Stokes equations 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 76D07 Stokes and related (Oseen, etc.) flows

NGSolve; Netgen
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### References:

 [1] M. Abramowitz, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1974. · Zbl 0171.38503 [2] G. Andrews, R. Askey, and R. Roy, Special functions, in Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1999, pp. 293–315. · Zbl 0920.33001 [3] S. Beuchler, V. Pillwein, and S. Zaglmayr, Sparsity optimized high order finite element functions for h(div) on simplices, Numer. Math., 122 (2012), pp. 197–225. · Zbl 1256.65100 [4] S. Beuchler and J. Schöberl, New shape functions for triangular $$p$$-FEM using integrated Jacobi polynomials, Numer. Math., 103 (2006), pp. 339–366. · Zbl 1095.65101 [5] D. Boffi, F. Brezzi, and M. Fortin, Mixed Finite Element Methods and Applications, Springer Science & Business Media, New York, 2013. · Zbl 1277.65092 [6] D. Braess, Finite Elemente—Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie, Springer, Berlin, 2013. [7] 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 [8] B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47 (2009), pp. 1319–1365. · Zbl 1205.65312 [9] B. Cockburn, J. Gopalakrishnan, N. Nguyen, J. Peraire, and F.-J. Sayas, Analysis of HDG methods for Stokes flow, Math. Comput., 80 (2011), pp. 723–760. · Zbl 1410.76164 [10] B. Cockburn, G. Kanschat, and D. Schötzau, A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comput., 74 (2005), pp. 1067–1095. · Zbl 1069.76029 [11] 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. [12] H. Egger and C. Waluga, hp-Analysis of a hybrid DG method for Stokes flow, IMA J. Numer. Anal., (2012), p. drs018. · Zbl 1328.76040 [13] G. Fu, Y. Jin, and W. Qiu, Parameter-Free Superconvergent $$H(\text{div})$$-Conforming HDG Methods for the Brinkman Equations, preprint, arXiv, , 2016. [14] J. Guzmán, C.-W. Shu, and F. A. Sequeira, H (div) conforming and DG methods for incompressible Eulers equations, IMA J. Numer. Anal., (2016), p. drw054. [15] P. Hansbo and M. G. Larson, Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method, Comp. Methods Appl. Mech. Eng., 191 (2002), pp. 1895–1908. · Zbl 1098.74693 [16] V. John, A. Linke, C. Merdon, M. Neilan, and L. G. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev., 59 (2017), pp. 492–544. · Zbl 1426.76275 [17] G. Kanschat and Y. Mao, Multigrid methods for Hdiv-conforming discontinuous Galerkin methods for the Stokes equations, J. Numer. Math., 23 (2015), pp. 51–66. · Zbl 1330.76071 [18] G. Kanschat and D. Schötzau, Energy norm a posteriori error estimation for divergence-free discontinuous Galerkin approximations of the Navier–Stokes equations, Int. J. Numer. Methods Fluids, 57 (2008), pp. 1093–1113. · Zbl 1140.76020 [19] J. Könnö and R. Stenberg, Numerical computations with H(div)-finite elements for the Brinkman problem, Comput. Geosci., 16 (2012), pp. 139–158. · Zbl 1348.76100 [20] P. Lederer, Pressure-Robust Discretizations for Navier–Stokes Equations: Divergence-Free Reconstruction for Taylor–Hood Elements and High Order Hybrid Discontinuous Galerkin Methods, master’s thesis, Vienna Technical University, Vienna, 2016. [21] P. Lederer and J. Schöberl, Polynomial Robust Stability Analysis for H(div)-Conforming Finite Elements for the Stokes Equations, preprint, arXiv, , 2016. [22] P. L. Lederer, C. Lehrenfeld, and J. Schöberl, Hybrid Discontinuous Galerkin Methods with Relaxed H(div)-Conformity for Incompressible Flows. Part I, preprint, arXiv, , 2018. [23] P. L. Lederer, A. Linke, C. Merdon, and J. Schöberl, Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements, SIAM J. Numer. Anal., 55 (2017), pp. 1291–1314. · Zbl 1457.65202 [24] C. Lehrenfeld, Hybrid Discontinuous Galerkin Methods for Solving Incompressible Flow Problems, Rheinisch-Westfalischen Technischen Hochschule, Aachen, 2010. [25] C. Lehrenfeld and J. Schöberl, High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows, Comp. Methods Appl. Mech. Eng., 307 (2016), pp. 339–361. [26] A. Linke, A divergence-free velocity reconstruction for incompressible flows, C. R. Math. Acad. Sci. Paris, 350 (2012), pp. 837–840. · Zbl 1303.76106 [27] A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime, Comp. Methods Appl. Mech. Eng., 268 (2014), pp. 782–800. · Zbl 1295.76007 [28] 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: M2AN, 50 (2016), pp. 289–309. · Zbl 1381.76186 [29] J. C. Nédélec, A new family of mixed finite elements in R3, Numer. Math., 50 (1986), pp. 57–81. [30] S. Rhebergen and G. Wells, Analysis of an Hybridized/Interface Stabilized Finite Element Method for the Stokes Equations, preprint, arXiv, , 2016. [31] J. Schöberl, NETGEN: An advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci., 1 (1997), pp. 41–52. · Zbl 0883.68130 [32] J. Schöberl, C++11 implementation of finite elements in NGSolve, Technical report ASC-2014-30, Institute for Analysis and Scientific Computing, TU Wien, Vienna, 2014. [33] P. W. Schroeder and G. Lube, Divergence-Free $$H({div})$$-FEM for Time-Dependent Incompressible Flows with Applications to High Reynolds Number Vortex Dynamics, preprint, arXiv, , 2017. · Zbl 1392.35210 [34] T. Warburton and J. Hesthaven, On the constants in HP-finite element trace inverse inequalities, Comput. Methods Appl. Mech. Eng., 192 (2003), pp. 2765–2773. · Zbl 1038.65116 [35] S. Zaglmayr, High order Finite Element Methods for Electromagnetic Field Computation, Ph.D. thesis, Johannes Kepler University, Linz, 2006.
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