Free Access
Volume 50, Number 1, January-February 2016
Page(s) 289 - 309
Published online 28 January 2016
  1. T. Apel and G. Matthies, Nonconforming, anisotropic, rectangular finite elements of arbitrary order for the Stokes problem. SIAM J. Numer. Anal. 46 (2008) 1867–1891. [CrossRef] [Google Scholar]
  2. D.N. Arnold and G. Awanou, Finite element differential forms on cubical meshes. Math. Comput. 83 (2014) 1551–1570. [Google Scholar]
  3. D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. Vol. 44 of Springer Ser. Comput. Math. Springer, Heidelberg (2013). [Google Scholar]
  4. M. Braack, E. Burman, V. John and G. Lube, Stabilized finite element methods for the generalized Oseen problem. Comput. Methods Appl. Mech. Engrg. 196 (2007) 853–866. [CrossRef] [MathSciNet] [Google Scholar]
  5. J.H. Bramble and S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 112–124. [Google Scholar]
  6. C. Brennecke, A. Linke, C. Merdon and J. Schöberl, Optimal and pressure-independent L2 velocity error estimates for a modified Crouzeix–Raviart Stokes element with BDM reconstructions. J. Comput. Math. 33 (2015) 191–208. [CrossRef] [Google Scholar]
  7. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Vol. 15 of Texts Appl. Math. 3rd edition. Springer, New York (2008). [Google Scholar]
  8. F. Brezzi, J. Douglas, Jr. and LD. Marini, Recent results on mixed finite element methods for second order elliptic problems. In Vistas in Applied Mathematics, Transl. Ser. Math. Engrg. Optimization Software, New York (1986). [Google Scholar]
  9. F. Brezzi, J. Douglas, Jr., R. Durán and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51 (1987) 237–250. [CrossRef] [MathSciNet] [Google Scholar]
  10. A. Buffa, C. de Falco and G. Sangalli, IsoGeometric Analysis: stable elements for the 2D Stokes equation. Int. J. Numer. Methods Fluids 65 (2011) 1407–1422. [CrossRef] [Google Scholar]
  11. 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. [CrossRef] [MathSciNet] [Google Scholar]
  12. M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973) 33–75. [Google Scholar]
  13. O. Dorok, W. Grambow and L. Tobiska, Aspects of finite element discretizations for solving the Boussinesq approximation of the Navier–Stokes Equations. Notes on Numerical Fluid Mechanics: Numerical Methods for the Navier–Stokes Equations 47 (1994) 50–61. [Google Scholar]
  14. C. Druzgalski, M. Andersen and A. Mani, Direct numerical simulation of electroconvective instability and hydrodynamic chaos near an ion-selective surface. Phys. Fluids 25 (2013). [Google Scholar]
  15. J.A. Evans and T.J.R. Hughes, Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations. Math. Models Methods Appl. Sci. 23 (2013) 1421–1478. [CrossRef] [Google Scholar]
  16. R.S. Falk and M. Neilan, Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51 (2013) 1308–1326. [CrossRef] [Google Scholar]
  17. M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 11 (1977) 341–354. [MathSciNet] [Google Scholar]
  18. L.P. Franca and T.J.R. Hughes, Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Engrg. 69 (1988) 89–129. [CrossRef] [MathSciNet] [Google Scholar]
  19. K.J. Galvin, A. Linke, L.G. Rebholz and N.E. Wilson, Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection. Comput. Methods Appl. Mech. Engrg. 237–240 (2012) 166–176. [CrossRef] [Google Scholar]
  20. S. Ganesan, G. Matthies and L. Tobiska, On spurious velocities in incompressible flow problems with interfaces. Comput. Methods Appl. Mech. Engrg. 196 (2007) 1193–1202. [CrossRef] [MathSciNet] [Google Scholar]
  21. J.-F. Gerbeau, C. Le Bris and M. Bercovier, Spurious velocities in the steady flow of an incompressible fluid subjected to external forces. Internat. J. Numer. Methods Fluids 25 (1997) 679–695. [CrossRef] [MathSciNet] [Google Scholar]
  22. V. Girault and P.-A. Raviart, Finite element methods for Navier-Stokes equations. Theory and algorithms. Vol. 5 of Springer Ser. Comput. Math. Springer-Verlag, Berlin (1986). [Google Scholar]
  23. J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements in three dimensions. IMA J. Numer. Anal. 34 (2014) 1489–1508,. [CrossRef] [MathSciNet] [Google Scholar]
  24. J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83 (2014) 15–36. [Google Scholar]
  25. J.P. Hennart, J. Jaffré and J.E. Roberts, A constructive method for deriving finite elements of nodal type. Numer. Math. 53 (1988) 701–738. [CrossRef] [MathSciNet] [Google Scholar]
  26. E.W. Jenkins, V. John, A. Linke and L.G. Rebholz, On the parameter choice in grad-div stabilization for the Stokes equations. Adv. Comput. Math. 40 (2014) 491–516. [CrossRef] [Google Scholar]
  27. G. Kanschat and N. Sharma, Divergence-conforming discontinuous Galerkin methods and C0 interior penalty methods. SIAM J. Numer. Anal. 52 (2014) 1822–1842. [CrossRef] [Google Scholar]
  28. M. Koddenbrock, Effizienz und Genauigkeit einer divergenzfreien Diskretisierung für die stationären inkompressiblen Navier–Stokes-Gleichungen. Master’s thesis, Freie Universität Berlin, Germany (2014). [Google Scholar]
  29. C. Lehrenfeld, Hybrid discontinuous Galerkin methods for incompressible flow problems. Master’s thesis, RWTH Aachen, Germany (2010). [Google Scholar]
  30. A. Linke, Collision in a cross-shaped domain – a steady 2d Navier–Stokes example demonstrating the importance of mass conservation in CFD. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3278–3286. [CrossRef] [MathSciNet] [Google Scholar]
  31. A. Linke, On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Engrg. 268 (2014) 782–800. [Google Scholar]
  32. G. Matthies, Inf-sup stable nonconforming finite elements of higher order on quadrilaterals and hexahedra. ESAIM: M2AN 41 (2007) 855–874. [CrossRef] [EDP Sciences] [Google Scholar]
  33. G. Matthies and L. Tobiska, The inf-sup condition for the mapped Qk-Formula element in arbitrary space dimensions. Computing 69 (2002) 119–139. [CrossRef] [MathSciNet] [Google Scholar]
  34. G. Matthies and L. Tobiska, Inf-sup stable non-conforming finite elements of arbitrary order on triangles. Numer. Math. 102 (2005) 293–309. [CrossRef] [MathSciNet] [Google Scholar]
  35. G. Matthies and L. Tobiska, Mass conservation of finite element methods for coupled flow-transport problems. Int. J. Comput. Sci. Math. 1 (2007) 293–307. [CrossRef] [MathSciNet] [Google Scholar]
  36. J.-C. Nédélec, Mixed finite elements in R3. Numer. Math. 35 (1980) 315–341. [CrossRef] [MathSciNet] [Google Scholar]
  37. M.A. Olshanskii, G. Lube, T. Heister and J. Löwe, Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 198 (2009) 3975–3988. [Google Scholar]
  38. M.A. Olshanskii and A. Reusken, Grad-div stabilization for Stokes equations. Math. Comput. 73 (2004) 1699–1718. [CrossRef] [Google Scholar]
  39. P.-A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems. In Mathematical aspects of finite element methods. Proc. Conf., Consiglio Naz. delle Ricerche, C.N.R., Rome, 1975. Vol. 606 of Lect. Notes Math. Springer, Berlin (1977) 292–315. [Google Scholar]
  40. H.-G. Roos, M. Stynes and L. Tobiska, Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems. Vol. 24 of Springer Ser. Comput. Math. 2nd edition. Springer-Verlag, Berlin (2008). [Google Scholar]
  41. L.R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modél. Math. Anal. Numér. 19 (1985) 111–143. [MathSciNet] [Google Scholar]
  42. J. Wang, Y. Wang and X. Ye, A posteriori error estimation for an interior penalty type method employing H(div) elements for the Stokes equations. SIAM J. Sci. Comput. 33 (2011) 131–152. [CrossRef] [Google Scholar]
  43. S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74 (2005) 543–554. [CrossRef] [Google Scholar]

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