Numerical Analysis of a BDF2 Modular Grad–Div Stabilization Method for the Navier–Stokes Equations

A second-order accurate modular algorithm is presented for a standard BDF2 code for the Navier–Stokes equations (NSE). The algorithm exhibits resistance to solver breakdown and increased computational efficiency for increasing values of grad–div parameters. We provide a complete theoretical analysis of the algorithms stability and convergency. Computational tests are performed and illustrate the theory and advantages over monolithic grad–div stabilizations.

[1]  M. Olshanskii,et al.  Stable finite‐element calculation of incompressible flows using the rotation form of convection , 2002 .

[2]  R. Rannacher,et al.  Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization , 1990 .

[3]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[4]  Volker John,et al.  Reference values for drag and lift of a two‐dimensional time‐dependent flow around a cylinder , 2004 .

[5]  M. Olshanskii.  A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a stabilization issue and iterative methods , 2002 .

[6]  R. Rannacher,et al.  Benchmark Computations of Laminar Flow Around a Cylinder , 1996 .

[7]  Leo G. Rebholz,et al.  Preconditioning sparse grad-div/augmented Lagrangian stabilized saddle point systems , 2013, Comput. Vis. Sci..

[8]  William Layton,et al.  An efficient and modular grad–div stabilization , 2017, Computer Methods in Applied Mechanics and Engineering.

[9]  W. Layton,et al.  On the determination of the grad-div criterion , 2017, Journal of Mathematical Analysis and Applications.

[10]  Leo G. Rebholz,et al.  On the convergence rate of grad-div stabilized Taylor–Hood to Scott–Vogelius solutions for incompressible flow problems , 2011 .

[11]  Jean-Luc Guermond,et al.  High-order time stepping for the Navier-Stokes equations with minimal computational complexity , 2016, J. Comput. Appl. Math..

[12]  Volker John,et al.  Analysis of the grad-div stabilization for the time-dependent Navier–Stokes equations with inf-sup stable finite elements , 2016, Adv. Comput. Math..

[13]  T. Hughes,et al.  Two classes of mixed finite element methods , 1988 .

[14]  G. Rapin,et al.  Efficient augmented Lagrangian‐type preconditioning for the Oseen problem using Grad‐Div stabilization , 2013 .

[15]  Maxim A. Olshanskii,et al.  On the accuracy of the rotation form in simulations of the Navier-Stokes equations , 2009, J. Comput. Phys..

[16]  Lutz Tobiska,et al.  A Two-Level Method with Backtracking for the Navier--Stokes Equations , 1998 .

[17]  Volker John,et al.  On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows , 2015, SIAM Rev..

[18]  Volker John,et al.  Time‐dependent flow across a step: the slip with friction boundary condition , 2006 .

[19]  Volker John,et al.  Grad-div Stabilization for the Evolutionary Oseen Problem with Inf-sup Stable Finite Elements , 2015, J. Sci. Comput..

[20]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[21]  Maxim A. Olshanskii,et al.  Grad–div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations , 2009 .

[22]  M. Stynes,et al.  Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems , 1996 .

[23]  Nikolaos A. Malamataris,et al.  Computer-aided analysis of flow past a surface-mounted obstacle , 1997 .

[24]  Maxim A. Olshanskii,et al.  Grad-div stablilization for Stokes equations , 2003, Math. Comput..

[25]  Michael McLaughlin,et al.  A conservative, second order, unconditionally stable artificial compression method , 2017 .

[26]  G. Burton.  Sobolev Spaces , 2013 .

[27]  P. E. Bernatz,et al.  How conservative? , 1971, The Annals of thoracic surgery.

[28]  Leo G. Rebholz,et al.  On a reduced sparsity stabilization of grad–div type for incompressible flow problems , 2013 .

[29]  Andreas Prohl,et al.  On Pressure Approximation via Projection Methods for Nonstationary Incompressible Navier-Stokes Equations , 2008, SIAM J. Numer. Anal..

[30]  W. Layton,et al.  On relaxation times in the Navier-Stokes-Voigt model , 2013 .

[31]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[32]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[33]  William Layton,et al.  Introduction to the Numerical Analysis of Incompressible Viscous Flows , 2008 .

[34]  Leo G. Rebholz,et al.  Error analysis and iterative solvers for Navier–Stokes projection methods with standard and sparse grad-div stabilization , 2014 .

[35]  Y. Rong,et al.  A partitioned second‐order method for magnetohydrodynamic flows at small magnetic reynolds numbers , 2017 .

[36]  Volker John,et al.  On the parameter choice in grad-div stabilization for the Stokes equations , 2014, Adv. Comput. Math..