Grad-div Stabilization for the Evolutionary Oseen Problem with Inf-sup Stable Finite Elements

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• 作者：Javier de Frutos ; Bosco García-Archilla ; Volker John…
• 关键词：Time ; dependent Oseen equations ; Inf ; sup stable pairs of finite element spaces ; Grad ; div stabilization ; Backward Euler scheme ; Two ; step backward differentiation scheme (BDF2) ; Crank–Nicolson scheme ; Uniform error estimates
• 刊名：Journal of Scientific Computing
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：66
• 期：3
• 页码：991-1024
• 全文大小：975 KB
• 参考文献：1.Ayuso, B., García-Archilla, B., Novo, J.: The postprocessed mixed finite-element method for the Navier–Stokes equations. SIAM J. Numer. Anal. 43(3), 1091–1111 (2005)CrossRef MathSciNet MATH
  2.Börm, S., Le Borne, S.: \(\cal H\) -LU factorization in preconditioners for augmented Lagrangian and grad-div stabilized saddle point systems. Int. J. Numer. Methods Fluids 68(1), 83–98 (2012)CrossRef MATH
  3.Bowers, A.L., Le Borne, S., Rebholz, L.G.: Error analysis and iterative solvers for Navier–Stokes projection methods with standard and sparse grad-div stabilization. Comput. Methods Appl. Mech. Eng. 275, 1–19 (2014)CrossRef MATH
  4.Braack, M., Burman, E., John, V., Lube, G.: Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Eng. 196(4–6), 853–866 (2007)CrossRef MathSciNet MATH
  5.Case, M.A., Ervin, V.J., Linke, A., Rebholz, L.G.: A connection between Scott–Vogelius and grad-div stabilized Taylor–Hood FE approximations of the Navier–Stokes equations. SIAM J. Numer. Anal. 49(4), 1461–1481 (2011)CrossRef MathSciNet MATH
  6.Ciarlet, Philippe G.: The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4
  7.Codina, R.: Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput. Methods Appl. Mech. Eng. 191(39–40), 4295–4321 (2002)CrossRef MathSciNet MATH
  8.Codina, R.: Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales. Appl. Numer. Math. 58(3), 264–283 (2008)CrossRef MathSciNet MATH
  9.Codina, R., Principe, J., Guasch, O., Badia, S.: Time dependent subscales in the stabilized finite element approximation of incompressible flow problems. Comput. Methods Appl. Mech. Eng. 196(21–24), 2413–2430 (2007)CrossRef MathSciNet MATH
  10.Dallmann, H., Arndt, D., Lube, G.: Local projection stabilization for the Oseen problem. Preprint 09, Institut für Numerische und Angewandte Mathematik, University of Göttingen, (2014)
  11.de Frutos, J., García-Archilla, B., John, V., Novo, J.: Grad-div stabilization for the time-dependent Navier–Stokes equations with inf-sup stable slements. In preparation, (2015)
  12.de Frutos, J., García-Archilla, B., Novo, J.: Local error estimates for the SUPG method applied to evolutionary convection–reaction–diffusion equations. J. Sci. Comput. (2015). doi:10.​1007/​s10915-015-0035-2
  13.Gelhard, T., Lube, G., Olshanskii, M.A., Starcke, J.-H.: Stabilized finite element schemes with LBB-stable elements for incompressible flows. J. Comput. Appl. Math. 177, 243–267 (2005)CrossRef MathSciNet MATH
  14.Girault, V., Raviart, P.A.: Finite element methods for Navier–Stokes equations, volume 5 of springer series in computational mathematics. Springer, Berlin (1986)
  15.Heister, T., Rapin, G.: Efficient augmented Lagrangian-type preconditioning for the Oseen problem using grad-div stabilization. Int. J. Numer. Methods Fluids 71(1), 118–134 (2013)CrossRef MathSciNet
  16.Jenkins, E.W., John, V., Linke, A., Rebholz, L.G.: On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40(2), 491–516 (2014)CrossRef MathSciNet MATH
  17.John, V., Kindl, A.: Numerical studies of finite element variational multiscale methods for turbulent flow simulations. Comput. Methods Appl. Mech. Eng. 199(1316), 841–852 (2010)CrossRef MathSciNet MATH
  18.John, V., Matthies, G.: MooNMD–a program package based on mapped finite element methods. Comput. Vis. Sci. 6(2–3), 163–169 (2004)CrossRef MathSciNet MATH
  19.Lube, G., Rapin, G.: Residual-based stabilized higher-order FEM for a generalized Oseen problem. Math. Models Methods Appl. Sci. 16, 949–966 (2006)CrossRef MathSciNet MATH
  20.Matthies, G., Lube, G., Röhe, L.: Some remarks on residual-based stabilisation of inf-sup stable discretisations of the generalised Oseen problem. Comput. Methods Appl. Math. 9, 368–390 (2009)CrossRef MathSciNet MATH
  21.Olshanskii, M., Lube, G., Heister, T., Löwe, J.: Grad-div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 198(49–52), 3975–3988 (2009)CrossRef MATH
  22.Olshanskii, M.A.: A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a stabilization issue and iterative methods. Comput. Methods Appl. Mech. Eng. 191(47–48), 5515–5536 (2002)CrossRef MathSciNet
  23.Olshanskii, M.A., Reusken, A.: Grad-div stabilization for Stokes equations. Math. Comput. 73(248), 1699–1718 (2004)CrossRef MathSciNet MATH
  24.Martin, R., Lutz, T.: Robust Numerical Methods for Singularly Perturbed Differential Equations, Volume 24 of Springer Series in Computational Mathematics. Convection–Diffusion–Reaction and Flow Problems, 2nd edn. Springer, Berlin (2008)
  25.Scott, L.R., Zhang, Shangyou: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)CrossRef MathSciNet MATH
  26.Tobiska, L., Verfürth, R.: Analysis of a streamline diffusion finite element method for the Stokes and Navier–Stokes equations. SIAM J. Numer. Anal. 33, 107–127 (1996)CrossRef MathSciNet MATH
  
• 作者单位：Javier de Frutos (1)
  Bosco García-Archilla (2)
  Volker John (3) (4)
  Julia Novo (5)
  
  1. Instituto de Investigación en Matemáticas (IMUVA), Universidad de Valladolid, Valladolid, Spain
  2. Departamento de Matemática Aplicada II, Universidad de Sevilla, Sevilla, Spain
  3. Weierstrass Institute for Applied Analysis and Stochastics, Leibniz Institute in Forschungsverbund Berlin e.V. (WIAS), Mohrenstr. 39, 10117, Berlin, Germany
  4. Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 6, 14195, Berlin, Germany
  5. Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691

文摘

The approximation of the time-dependent Oseen problem using inf-sup stable mixed finite elements in a Galerkin method with grad-div stabilization is studied. The main goal is to prove that adding a grad-div stabilization term to the Galerkin approximation has a stabilizing effect for small viscosity. Both the continuous-in-time and the fully discrete case (backward Euler method, the two-step BDF, and Crank–Nicolson schemes) are analyzed. In fact, error bounds are obtained that do not depend on the inverse of the viscosity in the case where the solution is sufficiently smooth. The bounds for the divergence of the velocity as well as for the pressure are optimal. The analysis is based on the use of a specific Stokes projection. Numerical studies support the analytical results. Keywords Time-dependent Oseen equations Inf-sup stable pairs of finite element spaces Grad-div stabilization Backward Euler scheme Two-step backward differentiation scheme (BDF2) Crank–Nicolson scheme Uniform error estimates