Advertisement

Computational Mechanics

, Volume 50, Issue 6, pp 667–674 | Cite as

Discrete spectrum analyses for various mixed discretizations of the Stokes eigenproblem

  • John A. EvansEmail author
  • Thomas J. R. Hughes
Original Paper

Abstract

We conduct discrete spectrum analyses for a selection of mixed discretization schemes for the Stokes eigenproblem. In particular, we consider the MINI element, the Crouzeix–Raviart element, the Marker-and-Cell scheme, the Taylor–Hood element, the \({\mathbf{Q}_{k}/P_{k-1}}\) element, the divergence-conforming discontinuous Galerkin method, and divergence-conforming B-splines. For each of these schemes, we compare the spectrum for the continuous Stokes problem with the spectrum for the discrete Stokes problem, and we discuss the relationship of eigenvalue errors with solution errors associated with unsteady viscous flow problems.

Keywords

Stokes eigenproblem Mixed methods Discrete spectrum analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold DN, Brezzi F, Fortin M (1984) A stable finite element for the Stokes equations. Calcolo 21: 337–344MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bazilevs Y, Calo VM, Cottrell JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197: 173–201MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bernardi C, Maday Y (1999) Uniform inf-sup conditions for the spectral discretization of the Stokes problem. Math Models Methods Appl Sci 9: 395–414MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Boffi D (2010) Finite element approximation of eigenvalue problems. Acta Numerica 19: 1–120MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Buffa A, de Falco C, Sangalli G (2011) Isogeometric analysis: stable elements for the 2D Stokes equation. Int J Numer Methods Fluids 65:1407–1422,20–30Google Scholar
  6. 6.
    Buffa A, Rivas J, Sangalli G, Vázquez R (2011) Isogeometric discrete differential forms in three dimensions. SIAM J Numer Anal 49: 818–844MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Buffa A, Sangalli G, Vázquez R (2010) Isogeometric analysis in electromagnetics: B-splines approximation. Comput Methods Appl Mech Eng 199: 1143–1152zbMATHCrossRefGoogle Scholar
  8. 8.
    Cockburn B, Kanschat G, Schötzau D (2004) A locally conservative LDG method for the incompressible Navier–Stokes equations. Math Comput 74: 1067–1095CrossRefGoogle Scholar
  9. 9.
    Cockburn B, Kanschat G, Schötzau D (2007) A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes equations. SIAM J Sci Comput 31: 61–73zbMATHGoogle Scholar
  10. 10.
    Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, ChichesterGoogle Scholar
  11. 11.
    Crouzeix M, Raviart PA (1973) Conforming and non-conforming finite element methods for solving the stationary Stokes equations. R.A.I.R.O. Anal Numérique 7: 33–76MathSciNetGoogle Scholar
  12. 12.
    Evans JA (2011) Divergence-free B-spline discretizations for viscous incompressible flows. Ph.D. thesis, The University of Texas at AustinGoogle Scholar
  13. 13.
    Evans JA, Hughes TJR (2012) Isogeometric divergence-conforming B-splines for the Darcy–Stokes–Brinkman equations. Math Models Methods Appl Sci doi: 10.1142/S0218202512500583
  14. 14.
    Evans JA, Hughes TJR (2012) Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations. Math Models Methods Appl Sci (In press)Google Scholar
  15. 15.
    Evans JA, Hughes TJR (2012) Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations. Tech Rep ICES Rep 12–16Google Scholar
  16. 16.
    Girault V, Lopez H (1996) Finite-element error estimates for the MAC scheme. IMA J Numer Anal 16: 347–379MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Harlow FH, Welch JE (1965) Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys Fluids 8: 2182zbMATHCrossRefGoogle Scholar
  18. 18.
    Hood P, Taylor C (1974) Navier–Stokes equations using mixed interpolation. In: Oden JT, Gallagher RH, Zienkiewicz OC, Taylor CFinite elementmethods in flowproblems..University ofAlabama in Huntsville Press, pp 121–132Google Scholar
  19. 19.
    Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, MineolazbMATHGoogle Scholar
  20. 20.
    Hughes TJR, Franca LP, Balestra M (1986) A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accomodating equal-order interpolations. Comput Methods Appl Mech Eng 59: 85–99MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hughes TJR, Reali A, Sangalli G (2008) Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p-method finite elements with k-method NURBS. Comput Methods Appl Mech Eng 197: 4104–4124MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Kanschat G (2008) Divergence-free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme. Int J Numer Methods Fluids 56: 941–950MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Matthies G, Tobiska L (2007) Mass conservation of finite element methods for coupled flow-transport problems. Int J Comput Sci Math 1: 293–307MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Nicolaides RA (1992) Analysis and convergence of the MAC scheme I. The linear problem. SIAM J Numer Anal 29: 1579–1591MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute for Computational Engineering and SciencesAustinUSA

Personalised recommendations