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Computing and Visualization in Science

, Volume 19, Issue 5–6, pp 47–63 | Cite as

Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?

  • Volker JohnEmail author
  • Petr Knobloch
  • Julia Novo
Special Issue FEM Symposium 2017

Abstract

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important open problems in these fields are discussed. The exposition concentrates on \(H^1\)-conforming finite elements.

Notes

Acknowledgements

The work of P. Knobloch was supported through the Grant No. 16-03230S of the Czech Science Foundation. The work of J. Novo was supported by Spanish MINECO under Grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER, UE). We would like to thank an anonymous referee whose suggestions helped us to improve this paper.

References

  1. 1.
    Acosta, G., Durán, R.G.: The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal. 37(1), 18–36 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ahmed, N., Bartsch, C., John, V., Wilbrandt, U.: An Assessment of Some Solvers for Saddle Point Problems Emerging from the Incompressible Navier–Stokes Equations. Comput. Methods Appl. Mech. Eng. 331, 492–513 (2018)MathSciNetGoogle Scholar
  3. 3.
    Ainsworth, M., Barrenechea, G.R., Wachtel, A.: Stabilization of high aspect ratio mixed finite elements for incompressible flow. SIAM J. Numer. Anal. 53(2), 1107–1120 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ainsworth, M., Coggins, P.: The stability of mixed \(hp\)-finite element methods for Stokes flow on high aspect ratio elements. SIAM J. Numer. Anal. 38(5), 1721–1761 (2000)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Allendes, A., Durán, F., Rankin, R.: Error estimation for low-order adaptive finite element approximations for fluid flow problems. IMA J. Numer. Anal. 36(4), 1715–1747 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Apel, T., Knopp, T., Lube, G.: Stabilized finite element methods with anisotropic mesh refinement for the Oseen problem. Appl. Numer. Math. 58(12), 1830–1843 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Apel, T., Randrianarivony, H.M.: Stability of discretizations of the Stokes problem on anisotropic meshes. Math. Comput. Simul. 61(3–6), 437–447 (2003)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Apel, T., Matthies, G.: Nonconforming, anisotropic, rectangular finite elements of arbitrary order for the Stokes problem. SIAM J. Numer. Anal. 46(4), 1867–1891 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Apel, T., Nicaise, S.: The inf-sup condition for low order elements on anisotropic meshes. Calcolo 41(2), 89–113 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Apel, T., Nicaise, S., Schöberl, J.: A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges. IMA J. Numer. Anal. 21(4), 843–856 (2001)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Arminjon, P., Dervieux, A.: Construction of TVD-like artificial viscosities on two-dimensional arbitrary FEM grids. J. Comput. Phys. 106(1), 176–198 (1993)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Arndt, D., Dallmann, H., Lube, G.: Local projection FEM stabilization for the time-dependent incompressible Navier–Stokes problem. Numer. Methods Part. Differ. Equ. 31(4), 1224–1250 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Augustin, M., Caiazzo, A., Fiebach, A., Fuhrmann, J., John, V., Linke, A., Umla, R.: An assessment of discretizations for convection-dominated convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 200(47–48), 3395–3409 (2011)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Babuška, I.: Error-bounds for finite element method. Numer. Math. 16, 322–333 (1971)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Bardos, C.W., Titi, E.S.: Mathematics and turbulence: where do we stand? J. Turbul. 14(3), 42–76 (2013)MathSciNetGoogle Scholar
  16. 16.
    Barrenechea, G.R., John, V., Knobloch, P.: A local projection stabilization finite element method with nonlinear crosswind diffusion for convection–diffusion–reaction equations. ESAIM Math. Model. Numer. Anal. 47(5), 1335–1366 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Barrenechea, G.R., John, V., Knobloch, P.: Some analytical results for an algebraic flux correction scheme for a steady convection–diffusion equation in one dimension. IMA J. Numer. Anal. 35(4), 1729–1756 (2015)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Barrenechea, G.R., John, V., Knobloch, P.: Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal. 54(4), 2427–2451 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Barrenechea, G.R., John, V., Knobloch, P.: An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes. Math. Models Methods Appl. Sci. 27(3), 525–548 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Barrenechea, G.R., Valentin, F.: Consistent local projection stabilized finite element methods. SIAM J. Numer. Anal. 48(5), 1801–1825 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Barrenechea, G.R., Valentin, F.: A residual local projection method for the Oseen equation. Comput. Methods Appl. Mech. Eng. 199(29–32), 1906–1921 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Barrenechea, G.R., Valentin, F.: Beyond pressure stabilization: a low-order local projection method for the Oseen equation. Int. J. Numer. Methods Eng. 86(7), 801–815 (2011)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Barrios, T.P., Cascón, J.M., González, M.: Augmented mixed finite element method for the Oseen problem: a priori and a posteriori error analyses. Comput. Methods Appl. Mech. Eng. 313, 216–238 (2017)MathSciNetGoogle Scholar
  24. 24.
    Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes. Math. Models Methods Appl. Sci. 16(7), 1031–1090 (2006)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Bazilevs, Y., Calo, V.M., Tezduyar, T.E., Hughes, T.J.R.: \(YZ\beta \) discontinuity capturing for advection-dominated processes with application to arterial drug delivery. Int. J. Numer. Methods Fluids 54(6–8), 593–608 (2007)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Becker, R., Braack, M.: A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4), 173–199 (2001)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Becker, R., Braack, M.: A two-level stabilization scheme for the Navier–Stokes equations. In: Feistauer, M., Dolejší, V., Knobloch, P., Najzar, K. (eds.) Numerical Mathematics and Advanced Applications, pp. 123–130. Springer, Berlin (2004)Google Scholar
  28. 28.
    Benzi, M., Olshanskii, M.A.: An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput. 28(6), 2095–2113 (2006)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Benzi, M., Wang, Z.: Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier–Stokes equations. SIAM J. Sci. Comput. 33(5), 2761–2784 (2011)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Berrone, S.: Robustness in a posteriori error analysis for FEM flow models. Numer. Math. 91(3), 389–422 (2002)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Bochev, P., Gunzburger, M.: An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations. SIAM J. Numer. Anal. 42(3), 1189–1207 (2004)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Boris, J.P., Book, D.L.: Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11(1), 38–69 (1973)zbMATHGoogle Scholar
  33. 33.
    Braack, M.: A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes. M2AN. Math. Model. Numer. Anal. 42(6), 903–924 (2008)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Braack, M., Burman, E., Taschenberger, N.: Duality based a posteriori error estimation for quasi-periodic solutions using time averages. SIAM J. Sci. Comput. 33(5), 2199–2216 (2011)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Braack, M., Lube, G., Röhe, L.: Divergence preserving interpolation on anisotropic quadrilateral meshes. Comput. Methods Appl. Math. 12(2), 123–138 (2012)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Braack, M., Mucha, P.B.: Directional do-nothing condition for the Navier-Stokes equations. J. Comput. Math. 32(5), 507–521 (2014)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Brennecke, C., Linke, A., Merdon, C., Schöberl, J.: Optimal and pressure-independent \(L^2\) velocity error estimates for a modified Crouzeix–Raviart Stokes element with BDM reconstructions. J. Comput. Math. 33(2), 191–208 (2015)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R–2), 129–151 (1974)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Brezzi, F., Fortin, M.: A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89(3), 457–491 (2001)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations. In: Efficient Solutions of Elliptic Systems (Kiel, 1984), Volume 10 of Notes Numer. Fluid Mech., pp. 11–19. Friedr. Vieweg, Braunschweig (1984)Google Scholar
  41. 41.
    Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Buffa, A., de Falco, C., Sangalli, G.: IsoGeometric analysis: stable elements for the 2D Stokes equation. Int. J. Numer. Methods Fluids 65(11–12), 1407–1422 (2011)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Bulling, J., John, V., Knobloch, P.: Isogeometric analysis for flows around a cylinder. Appl. Math. Lett. 63, 65–70 (2017)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Burman, E.: A posteriori error estimation for interior penalty finite element approximations of the advection–reaction equation. SIAM J. Numer. Anal. 47(5), 3584–3607 (2009)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Burman, E.: Robust error estimates for stabilized finite element approximations of the two dimensional Navier–Stokes’ equations at high Reynolds number. Comput. Methods Appl. Mech. Eng. 288, 2–23 (2015)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Burman, E., Ern, A.: Stabilized Galerkin approximation of convection–diffusion–reaction equations: discrete maximum principle and convergence. Math. Comput. 74(252), 1637–1652 (2005). (electronic)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Burman, E., Ern, A., Fernández, M.A.: Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem. ESAIM: M2AN 51(2), 487–507 (2017)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Burman, E., Fernández, M.A.: Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence. Numer. Math. 107(1), 39–77 (2007)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Burman, E., Guzmán, J., Leykekhman, D.: Weighted error estimates of the continuous interior penalty method for singularly perturbed problems. IMA J. Numer. Anal. 29(2), 284–314 (2009)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Burman, E., Hansbo, P.: Edge stabilization for Galerkin approximations of convection–diffusion–reaction problems. Comput. Methods Appl. Mech. Eng. 193(15–16), 1437–1453 (2004)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Burman, E., Hansbo, P.: Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Eng. 195(19–22), 2393–2410 (2006)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Burman, E., Santos, I.P.: Error estimates for transport problems with high Péclet number using a continuous dependence assumption. J. Comput. Appl. Math. 309, 267–286 (2017)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Charnyi, S., Heister, T., Olshanskii, M.A., Rebholz, L.G.: On conservation laws of Navier–Stokes Galerkin discretizations. J. Comput. Phys. 337, 289–308 (2017)MathSciNetGoogle Scholar
  54. 54.
    Chen, H.: Pointwise error estimates for finite element solutions of the Stokes problem. SIAM J. Numer. Anal. 44(1), 1–28 (2006)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Chizhonkov, E.V., Olshanskii, M.A.: On the domain geometry dependence of the LBB condition. M2AN Math. Model. Numer. Anal. 34(5), 935–951 (2000)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Codina, R., Blasco, J.: A finite element formulation for the Stokes problem allowing equal velocity–pressure interpolation. Comput. Methods Appl. Mech. Eng. 143(3–4), 373–391 (1997)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R–3), 33–75 (1973)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Dallmann, H., Arndt, D.: Stabilized finite element methods for the Oberbeck–Boussinesq model. J. Sci. Comput. 69(1), 244–273 (2016)MathSciNetzbMATHGoogle Scholar
  59. 59.
    de Frutos, J., García-Archilla, B., John, V., Novo, J.: An adaptive SUPG method for evolutionary convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 273, 219–237 (2014)MathSciNetzbMATHGoogle Scholar
  60. 60.
    de Frutos, J., García-Archilla, B., John, V., Novo, J.: Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements. Adv. Comput. Math. 44, 195–225 (2018)MathSciNetzbMATHGoogle Scholar
  61. 61.
    de Frutos, J., García-Archilla, B., John, V., Novo, J.: Error Analysis of Non Inf-sup Stable Discretizations of the time-dependent Navier–Stokes equations with Local Projection Stabilization. Technical Report arXiv:1709.01011 (2017)
  62. 62.
    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. 66(2), 528–554 (2016)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Dohrmann, C.R., Bochev, P.B.: A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Methods Fluids 46(2), 183–201 (2004)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Douglas Jr., J., Wang, J.P.: An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52(186), 495–508 (1989)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Du, S., Zhang, Z.: A robust residual-type a posteriori error estimator for convection–diffusion equations. J. Sci. Comput. 65(1), 138–170 (2015)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Durango, F., Novo, J.: Two-grid mixed finite-element approximations to the Navier-Stokes equations based on a Newton type-step. J. Sci. Comput. 74, 456–473 (2018)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Eigel, M., Merdon, C.: Equilibration a posteriori error estimation for convection–diffusion–reaction problems. J. Sci. Comput. 67(2), 747–768 (2016)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Elman, H., Howle, V.E., Shadid, J., Shuttleworth, R., Tuminaro, R.: Block preconditioners based on approximate commutators. SIAM J. Sci. Comput. 27(5), 1651–1668 (2006)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2014). Numerical Mathematics and Scientific ComputationzbMATHGoogle Scholar
  70. 70.
    Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations. Math. Models Methods Appl. Sci. 23(8), 1421–1478 (2013)MathSciNetzbMATHGoogle Scholar
  71. 71.
    Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations. J. Comput. Phys 241, 141–167 (2013)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Falk, R.S., Neilan, M.: Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51(2), 1308–1326 (2013)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Girault, V., Nochetto, R.H., Scott, L.R.: Max-norm estimates for Stokes and Navier–Stokes approximations in convex polyhedra. Numer. Math. 131(4), 771–822 (2015)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Girault, V., Raviart, P.-A.: Finite Element Approximation of the Navier–Stokes Equations, Volume 749 of Lecture Notes in Mathematics. Springer, Berlin (1979)Google Scholar
  75. 75.
    Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. Theory and algorithms. In: Volume 5 of Springer Series in Computational Mathematics. Springer, Berlin (1986)Google Scholar
  76. 76.
    Girault, V., Scott, L.R.: A quasi-local interpolation operator preserving the discrete divergence. Calcolo 40(1), 1–19 (2003)MathSciNetzbMATHGoogle Scholar
  77. 77.
    Glowinski, R.: Finite element methods for incompressible viscous flow. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. IX, pp. 3–1176. North-Holland, Amsterdam (2003)Google Scholar
  78. 78.
    Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.) 47(89), 271–306 (1959)MathSciNetzbMATHGoogle Scholar
  79. 79.
    Guzmán, J., Leykekhman, D.: Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra. Math. Comput. 81(280), 1879–1902 (2012)MathSciNetzbMATHGoogle Scholar
  80. 80.
    Guzmán, J., Neilan, M.: Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83(285), 15–36 (2014)MathSciNetzbMATHGoogle Scholar
  81. 81.
    Guzmán, J., Sánchez, M.A.: Max-norm stability of low order Taylor–Hood elements in three dimensions. J. Sci. Comput. 65(2), 598–621 (2015)MathSciNetzbMATHGoogle Scholar
  82. 82.
    Hauke, G., Doweidar, M.H., Fuster, D.: A posteriori error estimation for computational fluid dynamics: the variational multiscale approach. In: de Borst R., Ramm E. (eds) Multiscale Methods in Computational Mechanics, Lecture Notes in Applied and Computational Mechanics, vol. 55. Springer, Dordrecht (2010)Google Scholar
  83. 83.
    Hauke, G., Doweidar, M.H., Fuster, D., Gómez, A., Sayas, J.: Application of variational a-posteriori multiscale error estimation to higher-order elements. Comput. Mech. 38(4–5), 356–389 (2006)MathSciNetzbMATHGoogle Scholar
  84. 84.
    Hauke, G., Fuster, D., Doweidar, M.H.: Variational multiscale a-posteriori error estimation for multi-dimensional transport problems. Comput. Methods Appl. Mech. Eng. 197(33–40), 2701–2718 (2008)MathSciNetzbMATHGoogle Scholar
  85. 85.
    Hosseini, B.S., Möller, M., Turek, S.: Isogeometric analysis of the Navier–Stokes equations with Taylor–Hood B-spline elements. Appl. Math. Comput. 267, 264–281 (2015)MathSciNetzbMATHGoogle Scholar
  86. 86.
    Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)MathSciNetzbMATHGoogle Scholar
  87. 87.
    Hughes, T.J.R.: Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng. 127(1–4), 387–401 (1995)MathSciNetzbMATHGoogle Scholar
  88. 88.
    Hughes, T.J.R., Brooks, A.: A multidimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979), Volume 34 of AMD, pp. 19–35. Amer. Soc. Mech. Engrs. (ASME), New York (1979)Google Scholar
  89. 89.
    Hughes, T.J.R., Franca, L.P.: A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces. Comput. Methods Appl. Mech. Eng. 65(1), 85–96 (1987)MathSciNetzbMATHGoogle Scholar
  90. 90.
    Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška–Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59(1), 85–99 (1986)zbMATHGoogle Scholar
  91. 91.
    Hughes, T.J.R., Sangalli, G.: Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal. 45(2), 539–557 (2007)MathSciNetzbMATHGoogle Scholar
  92. 92.
    John, V.: A numerical study of a posteriori error estimators for convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 190(5–7), 757–781 (2000)MathSciNetzbMATHGoogle Scholar
  93. 93.
    John, V.: Finite element methods for incompressible flow problems, vol. 51 of Springer Series in Computational Mathematics. Springer, Cham (2016)Google Scholar
  94. 94.
    John, V., Kaiser, K., Novo, J.: Finite element methods for the incompressible Stokes equations with variable viscosity. ZAMM Z. Angew. Math. Mech. 96(2), 205–216 (2016)MathSciNetGoogle Scholar
  95. 95.
    John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations. I. A review. Comput. Methods Appl. Mech. Eng. 196(17–20), 2197–2215 (2007)MathSciNetzbMATHGoogle Scholar
  96. 96.
    John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. II. Analysis for \(P_1\) and \(Q_1\) finite elements. Comput. Methods Appl. Mech. Eng. 197(21–24), 1997–2014 (2008)zbMATHGoogle Scholar
  97. 97.
    John, V., Layton, W., Manica, C.C.: Convergence of time-averaged statistics of finite element approximations of the Navier–Stokes equations. SIAM J. Numer. Anal. 46(1), 151–179 (2007)MathSciNetzbMATHGoogle Scholar
  98. 98.
    John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59, 492–544 (2017)MathSciNetzbMATHGoogle Scholar
  99. 99.
    John, V., Mitkova, T., Roland, M., Sundmacher, K., Tobiska, L., Voigt, A.: Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Eng. Sci. 64(4), 733–741 (2009)Google Scholar
  100. 100.
    John, V., Novo, J.: On (essentially) non-oscillatory discretizations of evolutionary convection–diffusion equations. J. Comput. Phys. 231(4), 1570–1586 (2012)MathSciNetzbMATHGoogle Scholar
  101. 101.
    John, V., Novo, J.: A robust SUPG norm a posteriori error estimator for stationary convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 255, 289–305 (2013)MathSciNetzbMATHGoogle Scholar
  102. 102.
    John, V., Schmeyer, E.: Finite element methods for time-dependent convection–diffusion–reaction equations with small diffusion. Comput. Methods Appl. Mech. Eng. 198(3–4), 475–494 (2008)MathSciNetzbMATHGoogle Scholar
  103. 103.
    John, V., Schumacher, L.: A study of isogeometric analysis for scalar convection–diffusion equations. Appl. Math. Lett. 27, 43–48 (2014)MathSciNetzbMATHGoogle Scholar
  104. 104.
    Johnson, C., Schatz, A.H., Wahlbin, L.B.: Crosswind smear and pointwise errors in streamline diffusion finite element methods. Math. Comput. 49(179), 25–38 (1987)MathSciNetzbMATHGoogle Scholar
  105. 105.
    Knobloch, P.: Improvements of the Mizukami–Hughes method for convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 196(1–3), 579–594 (2006)MathSciNetzbMATHGoogle Scholar
  106. 106.
    Knopp, T., Lube, G., Rapin, G.: Stabilized finite element methods with shock capturing for advection–diffusion problems. Comput. Methods Appl. Mech. Eng. 191(27–28), 2997–3013 (2002)MathSciNetzbMATHGoogle Scholar
  107. 107.
    Kuzmin, D.: On the design of general-purpose flux limiters for finite element schemes. I. Scalar convection. J. Comput. Phys. 219(2), 513–531 (2006)MathSciNetzbMATHGoogle Scholar
  108. 108.
    Kuzmin, D.: Algebraic flux correction for finite element discretizations of coupled systems. In: Manolis, P., Eugenio, O., Bernard, S. (eds.) Proceedings of the International Conference on Computational Methods for Coupled Problems in Science and Engineering, pp. 1–5. CIMNE, Barcelona (2007)Google Scholar
  109. 109.
    Kuzmin, D.: Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. J. Comput. Appl. Math. 236(9), 2317–2337 (2012)MathSciNetzbMATHGoogle Scholar
  110. 110.
    Kuzmin, D., Möller, M.: Algebraic flux correction I. Scalar conservation laws. In: Kuzmin, D., Löhner, R., Turek, S. (eds.) Flux-Corrected Transport. Principles, Algorithms, and Applications, pp. 155–206. Springer, Berlin (2005)Google Scholar
  111. 111.
    Kuzmin, D., Turek, S.: High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. J. Comput. Phys. 198(1), 131–158 (2004)MathSciNetzbMATHGoogle Scholar
  112. 112.
    Layton, W.: Introduction to the Numerical Analysis of Incompressible Viscous Flows, Volume 6 of Computational Science & Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)Google Scholar
  113. 113.
    Lederer, P.L., Linke, A., Merdon, C., Schöberl, J.: Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements. SIAM J. Numer. Anal. 55(3), 1291–1314 (2017)MathSciNetzbMATHGoogle Scholar
  114. 114.
    Liao, Q., Silvester, D.: Robust stabilized Stokes approximation methods for highly stretched grids. IMA J. Numer. Anal. 33(2), 413–431 (2013)MathSciNetzbMATHGoogle Scholar
  115. 115.
    Linke, A.: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Eng. 268, 782–800 (2014)MathSciNetzbMATHGoogle Scholar
  116. 116.
    Linke, A., Matthies, G., Tobiska, L.: Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors. ESAIM Math. Model. Numer. Anal. 50(1), 289–309 (2016)MathSciNetzbMATHGoogle Scholar
  117. 117.
    Linke, A., Merdon, C.: Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 311, 304–326 (2016)MathSciNetGoogle Scholar
  118. 118.
    Löhner, R., Morgan, K., Peraire, J., Vahdati, M.: Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier–Stokes equations. Int. J. Numer. Methods Fluids 7(10), 1093–1109 (1987)zbMATHGoogle Scholar
  119. 119.
    Lube, G., Arndt, D., Dallmann, H.: Understanding the limits of inf-sup stable Galerkin-FEM for incompressible flows. In: Boundary and interior layers, computational and asymptotic methods—BAIL 2014, volume 108 of Lect. Notes Comput. Sci. Eng., pp. 147–169. Springer, Cham (2015)Google Scholar
  120. 120.
    Lube, G., Rapin, G.: Residual-based stabilized higher-order FEM for advection-dominated problems. Comput. Methods Appl. Mech. Eng. 195(33–36), 4124–4138 (2006)MathSciNetzbMATHGoogle Scholar
  121. 121.
    Micheletti, S., Perotto, S., Picasso, M.: Stabilized finite elements on anisotropic meshes: a priori error estimates for the advection–diffusion and the Stokes problems. SIAM J. Numer. Anal. 41(3), 1131–1162 (2003)MathSciNetzbMATHGoogle Scholar
  122. 122.
    Mizukami, A., Hughes, T.J.R.: A Petrov–Galerkin finite element method for convection-dominated flows: an accurate upwinding technique for satisfying the maximum principle. Comput. Methods Appl. Mech. Eng. 50(2), 181–193 (1985)MathSciNetzbMATHGoogle Scholar
  123. 123.
    Nävert, U.: A finite element method for convection–diffusion problems. Ph.D. Thesis, Chalmers University of Technology (1982)Google Scholar
  124. 124.
    Niijima, K.: Pointwise error estimates for a streamline diffusion finite element scheme. Numer. Math. 56(7), 707–719 (1990)MathSciNetzbMATHGoogle Scholar
  125. 125.
    Roos, H.-G., Stynes, M.: Some open questions in the numerical analysis of singularly perturbed differential equations. Comput. Methods Appl. Math. 15(4), 531–550 (2015)MathSciNetzbMATHGoogle Scholar
  126. 126.
    Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection–Diffusion and Flow Problems, vol. 24 of Springer Series in Computational Mathematics. Springer, Berlin (1996)Google Scholar
  127. 127.
    Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems, vol. 24 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2008)Google Scholar
  128. 128.
    Saad, Y.: A flexible inner–outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)MathSciNetzbMATHGoogle Scholar
  129. 129.
    Sangalli, G.: Robust a-posteriori estimator for advection–diffusion–reaction problems. Math. Comput. 77(261), 41–70 (2008). (electronic)MathSciNetzbMATHGoogle Scholar
  130. 130.
    Schötzau, D., Schwab, C., Stenberg, R.: Mixed \(hp\)-FEM on anisotropic meshes. II. Hanging nodes and tensor products of boundary layer meshes. Numer. Math. 83(4), 667–697 (1999)MathSciNetzbMATHGoogle Scholar
  131. 131.
    Schroeder, P.W., Lube, G.: Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier-Stokes flows. J. Num. Math., Accepted for publication (2017)Google Scholar
  132. 132.
    Schwegler, K., Bause, M.: Goal-oriented a posteriori error control for nonstationary convection-dominated transport problems. Technical Report arXiv:1601.06544 (2016)
  133. 133.
    Scott, L.R., Vogelius, M.: Conforming finite element methods for incompressible and nearly incompressible continua. In: Large-Scale Computations in Fluid Mechanics, Part 2 (La Jolla, Calif., 1983), Volume 22 of Lectures in Appl. Math., pp. 221–244. Amer. Math. Soc., Providence (1985)Google Scholar
  134. 134.
    Speleers, H., Manni, C., Pelosi, F., Sampoli, M.L.: Isogeometric analysis with Powell–Sabin splines for advection–diffusion–reaction problems. Comput. Methods Appl. Mech. Eng. 221/222, 132–148 (2012)MathSciNetzbMATHGoogle Scholar
  135. 135.
    Tabata, M., Tagami, D.: Error estimates for finite element approximations of drag and lift in nonstationary Navier–Stokes flows. Japan J. Ind. Appl. Math. 17(3), 371–389 (2000)MathSciNetzbMATHGoogle Scholar
  136. 136.
    Tobiska, L., Verfürth, R.: Robust a posteriori error estimates for stabilized finite element methods. IMA J. Numer. Anal. 35(4), 1652–1671 (2015)MathSciNetzbMATHGoogle Scholar
  137. 137.
    Vanka, S.P.: Block-implicit multigrid solution of Navier–Stokes equations in primitive variables. J. Comput. Phys. 65(1), 138–158 (1986)MathSciNetzbMATHGoogle Scholar
  138. 138.
    Verfürth, R.: A posteriori error estimators for convection–diffusion equations. Numer. Math. 80(4), 641–663 (1998)MathSciNetzbMATHGoogle Scholar
  139. 139.
    Verfürth, R.: Robust a posteriori error estimates for stationary convection–diffusion equations. SIAM J. Numer. Anal. 43(4), 1766–1782 (2005). (electronic)MathSciNetzbMATHGoogle Scholar
  140. 140.
    Wilbrandt, U., Bartsch, C., Ahmed, N., Alia, N., Anker, F., Blank, L., Caiazzo, A., Ganesan, S., Giere, S., Matthies, G., Meesala, R., Shamim, A., Venkatesan, J., John, V.: ParMooN—a modernized program package based on mapped finite elements. Comput. Math. Appl. 74(1), 74–88 (2017)MathSciNetzbMATHGoogle Scholar
  141. 141.
    Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31(3), 335–362 (1979)MathSciNetzbMATHGoogle Scholar
  142. 142.
    Zhang, S.: A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74(250), 543–554 (2005)MathSciNetzbMATHGoogle Scholar
  143. 143.
    Zhou, G.H., Rannacher, R.: Pointwise superconvergence of the streamline diffusion finite-element method. Numer. Methods Partial Differ. Equ 12(1), 123–145 (1996)MathSciNetzbMATHGoogle Scholar
  144. 144.
    Zhou, G.: How accurate is the streamline diffusion finite element method? Math. Comput. 66(217), 31–44 (1997)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsLeibniz Institute in Forschungsverbund Berlin e. V. (WIAS)BerlinGermany
  2. 2.Department of Mathematics and Computer ScienceFreie Universität BerlinBerlinGermany
  3. 3.Department of Numerical Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic
  4. 4.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain

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