## Hybrid discontinuous Galerkin methods with relaxed.(English)Zbl 1434.35058

The paper under consideration is the second part of a pair of papers considering Hybrid Discontinuous Galerkin methods with relaxed $$H(\operatorname{div})$$-conformity. The main objective of this second part consists in the presentation of a high order polynomial robust analysis for the relaxed $$H(\operatorname{div})$$-conforming Hybrid Discontinuous Galerkin discretization of the two dimensional Stokes problem. The base for this is given by the polynomial robust LBB-condition for BDM elements proven recently by two of the authors. It is derived by a direct approach instead of using a best approximation result. Furthermore, the impact of the reconstruction operator on the $$hp$$ analysis is studied and numerical examples for the polynomial robustness are considered, too. Finally, the authors present an efficient operator splitting time integration scheme fior the Navier-Stokes equations which includes the ideas of the reconstruction operator. The paper is rather comprehensive and self-contained. The bibliography contains 31 items.

### 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

### Software:

NGSolve; Netgen; FEATFLOW
Full Text:

### References:

 [1] I. Babuška, A. Craig, J. Mandel and J. Pitkäranta, Efficient preconditioning for the p-version finite element method in two dimensions. SIAM J. Numer. Anal. 28 (1991) 624-661. [2] I. Babuška and M. Suri, The hp version of the finite element method with quasiuniform meshes. RAIRO Model. Math. Anal. Numer. 21 (1987) 199-238. · Zbl 0623.65113 [3] J. Carrero, B. Cockburn and D. Schötzau, Hybridized globally divergence-free LDG methods. Part I: the Stokes problem. Math. Comput. 75 (2006) 533-563. · Zbl 1087.76061 [4] A. Chernov, Optimal convergence estimates for the trace of the polynomial L^2-projection operator on a simplex. Math. Comput. 81 (2012) 765-787. · Zbl 1242.41007 [5] B. Cockburn and J. Gopalakrishnan, Incompressible finite elements via hybridization. Part I: the Stokes system in two space dimensions. SIAM J. Numer. Anal. 43 (2005) 1627-1650. · Zbl 1145.76402 [6] B. Cockburn and J. Gopalakrishnan, Incompressible finite elements via hybridization. Part II: the Stokes system in three space dimensions. SIAM J. Numer. Anal. 43 (2005) 1651-1672. · Zbl 1145.76403 [7] B. Cockburn, J. Gopalakrishnan, N. Nguyen, J. Peraire and F.-J. Sayas, Analysis of HDG methods for Stokes flow. Math. Comput. 80 (2011) 723-760. · Zbl 1410.76164 [8] B. Cockburn, G. Kanschat and D. Schötzau, A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comput. 74 (2005) 1067-1095. · Zbl 1069.76029 [9] 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) 61-73. [10] D.A. Di Pietro, A. Ern, A. Linke and F. Schieweck, A discontinuous skeletal method for the viscosity-dependent stokes problem. Comput. Methods Appl. Mech. Eng. 306 (2016) 175-195. [11] H. Egger and C. Waluga, hp analysis of a hybrid DG method for Stokes flow. IMA J. Numer. Anal. 33 (2012) 687-721. · Zbl 1328.76040 [12] FEATFLOW Finite element software for the incompressible Navier-Stokes equations. Available at: (2019). [13] G. Fu, A high-order HDG method for the Biot’s consolidation model. Preprint arXiv: 1804.10329 (2018). [14] G. Fu, Y. Jin and W. QiuParameter-free superconvergent h(div)-conforming hdg methods for the brinkman equations. Preprint arXiv: 1607.07662 (2016). [15] G. Fu and C. Lehrenfeld, A strongly conservative hybrid DG/mixed FEM for the coupling of Stokes and Darcy flow. J. Sci. Comput. 77 (2018) 1605-1620. · Zbl 1406.65112 [16] V. John, A. Linke, C. Merdon, M. Neilan and L. Rebholz, On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59 (2017) 492-544. · Zbl 1426.76275 [17] L.I.G. Kovasznay, Laminar flow behind a two-dimensional grid. Math. Proc. Camb. Philos. Soc. 44 (1948) 58-62. · Zbl 0030.22902 [18] P.L. Lederer, C. Lehrenfeld and J. SchöberlHybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part I. Preprint arXiv: 1707.02782 (2017). [19] P.L. Lederer and J. Schöberl, Polynomial robust stability analysis for H(div)-conforming finite elements for the Stokes equations. IMA J. Numer. Anal. 38 (2018) 1832-1860. · Zbl 1462.65192 [20] C. Lehrenfeld and J. Schöberl, High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput. Methods Appl. Mech. Eng. 307 (2016) 339-361. [21] A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Eng. 268 (2014) 782-800. · Zbl 1295.76007 [22] A.J. Majda and A.L. Bertozzi, Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002). [23] J.M. Melenk and T. Wurzer, On the stability of the boundary trace of the polynomial L^2-projection on triangles and tetrahedra. Comput. Math. Appl. 67 (2014) 944-965. · Zbl 1350.65014 [24] J.M. Melenk and T. Apel, Interpolation and quasi-interpolation in h- and hp-version finite element spaces (extended version). ASC Report - Institute for analysis and Scientific Computing - Vienna University of Technology (2015) 39. [25] S. Rhebergen and G.N. Wells, Analysis of a hybridized/interface stabilized finite element method for the stokes equations. SIAM J. Numer. Anal. 55 (2017) 1982-2003. · Zbl 1426.76299 [26] M. Schäfer, S. Turek, F. Durst, E. Krause and R. RannacherBenchmark computations of laminar flow around a cylinder, in Flow Simulation with High-Performance Computers II, edited by E.H. by Hirschel. In Vol. 48 of Notes on Numerical Fluid Mechanics (NNFM). Springer Vieweg+Teubner Verlag (1996). [27] J. Schöberl, NETGEN An advancing front 2D/3D-mesh generator based on abstract rules. Comput. Visual. Sci. 1 (1997) 41-52. · Zbl 0883.68130 [28] J. Schöberl, C++11 implementation of finite elements in NGSolve. Technical Report ASC-2014-30, Institute for Analysis and Scientific Computing (September 2014). [29] P.W. Schroeder and G. Lube, Divergence-free H(div)-fem for time-dependent incompressible flows with applications to high Reynolds number vortex dynamics. J. Sci. Comput. 75 (2018) 830-858. · Zbl 1392.35210 [30] C. Schwab, p-and hp-finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Oxford University Press, Oxford (1998). · Zbl 0910.73003 [31] B. Stamm and T.P. Wihler, hp-optimal discontinuous Galerkin methods for linear elliptic problems. Math. Comp. 79 (2010) 2117-2133. · Zbl 1202.65151
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.