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Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

Philipp W. Schroeder, Christoph Lehrenfeld, Alexander Linke, Gert Lube
2018 SeMA Journal  
Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense.  ...  Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability.  ...  Grad-div Non-div-free grad-div stabilised Taylor-Hood, div-free Scott-Vogelius FEM and div-free BDM-(H)DG show the theoretical qualitative behaviour of the exponential Gronwall factor.  ... 
doi:10.1007/s40324-018-0157-1 fatcat:k7fvbdjqkzeixfujbl3clvfsya

On an Efficient Finite Element Method for Navier-Stokes-ω with Strong Mass Conservation

Carolina C. Manica, Monika Neda, Maxim Olshanskii, Leo G. Rebholz, Nicholas E. Wilson
2011 Computational Methods in Applied Mathematics  
We study an efficient finite element method for the NS-ω model, that uses van Cittert approximate deconvolution to improve accuracy and Scott-Vogelius elements to provide pointwise mass conservative solutions  ...  We provide a complete numerical analysis of the method, including well-posedness, unconditional stability, and optimal convergence.  ...  Scott-Vogelius and Taylor-Hood solutions in the above experiment of flow around a cylinder.  ... 
doi:10.2478/cmam-2011-0001 fatcat:rnqypmdb7ba6xnb2krskebmpce

On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond

Nicolas Gauger, Alexander Linke, Philipp Schroeder
2019 SMAI Journal of Computational Mathematics  
problem, based on discretisations with Scott-Vogelius SV k and Taylor-Hood TH k of different order k ∈ {4, 8}.  ...  In order to do so, we repeat the experiment presented in Figure 8 .1 now for the pressure-robust, divergence-free Scott-Vogelius method SV k and the non-pressure-robust Taylor-Hood element TH k ; cf.  ...  Note that the friction term −ν∆u is indeed a divergence-free contribution to the Navier-Stokes material derivative, which can be strong in the vicinity of the no-slip boundary of the domain.  ... 
doi:10.5802/smai-jcm.44 fatcat:22bfdcd2abdgbh535vaaiyejuu

Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier–Stokes flows

Philipp W. Schroeder, Gert Lube
2017 Journal of Numerical Mathematics  
AbstractThis article focusses on the analysis of a conforming finite element method for the time-dependent incompressible NavierStokes equations.  ...  For divergence-free approximations, in a semi-discrete formulation, we prove error estimates for the velocity that hold independently of both pressure and Reynolds number.  ...  Acknowledgements We gratefully acknowledge the comments and suggestions about this manuscript from the anonymous reviewers.  ... 
doi:10.1515/jnma-2016-1101 fatcat:a5hq2ubi5facdmcv7whd4vrhay

Divergence-Free H(div)-FEM for Time-Dependent Incompressible Flows with Applications to High Reynolds Number Vortex Dynamics

Philipp W. Schroeder, Gert Lube
2017 Journal of Scientific Computing  
Complementing the theoretical results, H(div)-FEM are applied to the simulation of full nonlinear Navier-Stokes problems.  ...  For convection-dominated problems, the proposed method relies on a velocity jump upwind stabilisation which is not gradient-based.  ...  Abbreviations of different methods: Non-div-free H H H 1 EMAC Taylor-Hood (eTHk) and equal-order (EOk), div-free H H H 1 Scott-Vogelius (SVk) and div-free H H H(div) Raviart-Thomas (RTk).  ... 
doi:10.1007/s10915-017-0561-1 fatcat:ev57wooebvbpnladvfqr6dljzm

Weierstraß-Institut für Angewandte Analysis und Stochastik Improving mass conservation in FE approximations of the Navier Stokes equations using continuous velocity fields: A connection between grad-div stabilization and Scott-Vogelius elements

Im Forschungsverbund, Berlin Preprint, M Case, V Ervin, A Linke, L Rebholz
2010 unpublished
We also prove that, again under these mild restrictions, the limit of the grad-div stabilized Taylor-Hood solutions to the Navier-Stokes problem converges to the Scott-Vogelius solution as the stabilization  ...  Numerical tests are provided which verify the theory, and show how both Scott-Vogelius and grad-div stabilized Taylor-Hood (with large stabilization parameter) elements can provide accurate results with  ...  Relationship between the Taylor-Hood and the Scott-Vogelius element Section 2 shows the Taylor-Hood and the Scott-Vogelius element are not unrelated to each other, as they differ only in their pressure  ... 
fatcat:63zkkdalsjfdxmdw5eo23mtxqm

Weierstraß-Institut für Angewandte Analysis und Stochastik Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations

Philipp Schroeder, Christoph Lehrenfeld, Alexander Linke, Gerd Lube, Philipp Schroeder, Christoph Lehrenfeld, Alexander Linke, Gerd Lube
2010 Mathematics Subject Classification. 35Q30, 65M15, 65M60, 76D17   unpublished
Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense.  ...  Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability.  ...  Non-div-free grad-div stabilised Taylor-Hood, div-free Scott-Vogelius FEM and div-free BDM-(H)DG show the theoretical qualitative behaviour of the H H H(div)-HDG method on the coarse mesh is shown with  ... 
fatcat:5caeahb3hbhrlm6gccxpyctb6y

Numerical and Analytical Aspects of POD-Based Reduced-Order Modeling in Computational Fluid Dynamics [article]

Swetlana Giere, Universitätsbibliothek Der FU Berlin, Universitätsbibliothek Der FU Berlin
2016
All investigations involve either convection-diffusion-reaction equation or the incompressible Navier-Stokes equations. The main contribution of the thesis can be divided into three parts.  ...  One of the choices for the stabilization parameter is recommended.  ...  A popular residual-based stabilization is the SUPG/PSPG/grad-div method, see [21] and the references therein.  ... 
doi:10.17169/refubium-17752 fatcat:sgahutxbrfaivc4vcga4ofegq4