The Stokes Equations

  • Volker John
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 51)


The Stokes equations model the simplest incompressible flow problems. These problems are steady-state and the convective term can be neglected. Hence, the arising model is linear. Thus, the only difficulty which remains from the problems mentioned in Remark  2.19 is the coupling of velocity and pressure.


Stokes Equations Convective Terms Incompressible Flow pressure-stabilizing/Petrov – Galerkin (PSPG) Finite Element Error Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Acosta G, Durán RG (1999) The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J Numer Anal 37:18–36 (electronic)Google Scholar
  2. Adams RA (1975) Sobolev spaces. Pure and applied mathematics, vol 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York/London, pp xviii+268Google Scholar
  3. Ainsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. Pure and applied mathematics (New York). Wiley-Interscience [Wiley], New York, pp xx+240Google Scholar
  4. Arnold D, Qin J (1992) Quadratic velocity/linear pressure Stokes elements. In: Vichnevetsky R, Knight D, Richter G (eds) Advances in computer methods for partial differential equations VII. IMACS, New Brunswick, pp 28–34Google Scholar
  5. Aubin J-P (1967) Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Gelerkin’s and finite difference methods. Ann Scuola Norm Sup Pisa (3) 21:599–637Google Scholar
  6. Barrenechea GR, Valentin F (2010a) Consistent local projection stabilized finite element methods. SIAM J Numer Anal 48:1801–1825MathSciNetCrossRefzbMATHGoogle Scholar
  7. Barrenechea GR, Valentin F (2010b) A residual local projection method for the Oseen equation. Comput Methods Appl Mech Eng 199:1906–1921MathSciNetCrossRefzbMATHGoogle Scholar
  8. Barrenechea GR, Valentin F (2011) Beyond pressure stabilization: a low-order local projection method for the Oseen equation. Int J Numer Methods Eng 86:801–815MathSciNetCrossRefzbMATHGoogle Scholar
  9. Barth T, Bochev P, Gunzburger M, Shadid J (2004) A taxonomy of consistently stabilized finite element methods for the Stokes problem. SIAM J Sci Comput 25:1585–1607MathSciNetCrossRefzbMATHGoogle Scholar
  10. Becker R, Braack M (2001) A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38:173–199MathSciNetCrossRefzbMATHGoogle Scholar
  11. Bochev P, Gunzburger M (2004) An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations. SIAM J Numer Anal 42:1189–1207MathSciNetCrossRefzbMATHGoogle Scholar
  12. Bochev PB, Dohrmann CR, Gunzburger MD (2006) Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J Numer Anal 44:82–101 (electronic)Google Scholar
  13. Boffi D, Brezzi F, Fortin M (2013) Mixed finite element methods and applications. Springer series in computational mathematics, vol 44. Springer, Heidelberg, pp xiv+685Google Scholar
  14. Bowers AL, Le Borne S, Rebholz LG (2014) Error analysis and iterative solvers for Navier-Stokes projection methods with standard and sparse grad-div stabilization. Comput Methods Appl Mech Eng 275:1–19MathSciNetCrossRefzbMATHGoogle Scholar
  15. Brennecke C, Linke A, Merdon C, Schöberl J (2015) Optimal and pressure-independent L 2 velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions. J Comput Math 33:191–208MathSciNetCrossRefzbMATHGoogle Scholar
  16. Brenner SC (2003) Poincaré-Friedrichs inequalities for piecewise H 1 functions. SIAM J Numer Anal 41:306–324MathSciNetCrossRefzbMATHGoogle Scholar
  17. Brezzi F, Douglas J Jr (1988) Stabilized mixed methods for the Stokes problem. Numer Math 53:225–235MathSciNetCrossRefzbMATHGoogle Scholar
  18. Brezzi F, Fortin M (2001) A minimal stabilisation procedure for mixed finite element methods. Numer Math 89:457–491MathSciNetCrossRefzbMATHGoogle Scholar
  19. Brezzi F, Pitkäranta J (1984) On the stabilization of finite element approximations of the Stokes equations. In: Efficient solutions of elliptic systems (Kiel, 1984). Notes Numerical Fluid Mechanics, vol 10. Friedr. Vieweg, Braunschweig, pp 11–19Google Scholar
  20. Bychenkov YV, Chizonkov EV (1999) Optimization of one three-parameter method of solving an algebraic system of the Stokes type. Russ J Numer Anal Math Modell 14:429–440MathSciNetCrossRefzbMATHGoogle Scholar
  21. Case MA, Ervin VJ, Linke A, Rebholz LG (2011) A connection between Scott-Vogelius and grad-div stabilized Taylor-Hood FE approximations of the Navier-Stokes equations. SIAM J Numer Anal 49:1461–1481MathSciNetCrossRefzbMATHGoogle Scholar
  22. Ciarlet PG (1978) The finite element method for elliptic problems. Studies in mathematics and its applications, vol 4. North-Holland Publishing Co., Amsterdam, pp xix+530Google Scholar
  23. Clough R, Tocher J (1965) Finite element stiffness matrices for analysis of plates in bending. In: Proceedings of the conference on matrix methods in structural mechanics. Wright Patterson A.F.B., OH, pp 515–545Google Scholar
  24. Codina R, Blasco J (1997) A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation. Comput Methods Appl Mech Eng 143:373–391MathSciNetCrossRefzbMATHGoogle Scholar
  25. Codina R, Blasco J (2000) Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations. Numer Math 87:59–81MathSciNetCrossRefzbMATHGoogle Scholar
  26. Dari E, Durán R, Padra C (1995) Error estimators for nonconforming finite element approximations of the Stokes problem. Math Comp 64:1017–1033MathSciNetCrossRefzbMATHGoogle Scholar
  27. Dauge M (1989) Stationary Stokes and Navier-Stokes systems on two- or three-dimensional domains with corners. I. Linearized equations. SIAM J Math Anal 20:74–97MathSciNetCrossRefzbMATHGoogle Scholar
  28. Dohrmann CR, Bochev PB (2004) A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int J Numer Methods Fluids 46:183–201MathSciNetCrossRefzbMATHGoogle Scholar
  29. Douglas J Jr, Wang JP (1989) An absolutely stabilized finite element method for the Stokes problem. Math Comp 52:495–508MathSciNetCrossRefzbMATHGoogle Scholar
  30. Ern A, Guermond J-L (2004) Theory and practice of finite elements. Applied mathematical sciences, vol 159. Springer, New York, pp xiv+524Google Scholar
  31. Evans LC (2010) Partial differential equations. Graduate studies in mathematics, vol 19, 2nd edn. American Mathematical Society, Providence, RI, pp xxii+749Google Scholar
  32. Falk RS, Neilan M (2013) Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J Numer Anal 51:1308–1326MathSciNetCrossRefzbMATHGoogle Scholar
  33. Franca LP, Hughes TJR (1988) Two classes of mixed finite element methods. Comput Methods Appl Mech Eng 69:89–129MathSciNetCrossRefzbMATHGoogle Scholar
  34. Franca LP, Hughes TJR, Stenberg R (1993) Stabilized finite element methods. In: Incompressible computational fluid dynamics: trends and advances. Cambridge University Press, Cambridge, pp 87–107CrossRefGoogle Scholar
  35. Galvin KJ, Linke A, Rebholz LG, Wilson NE (2012) Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection. Comput Methods Appl Mech Eng 237/240:166–176MathSciNetCrossRefzbMATHGoogle Scholar
  36. Gelhard T, Lube G, Olshanskii MA, Starcke J-H (2005) Stabilized finite element schemes with LBB-stable elements for incompressible flows. J Comput Appl Math 177:243–267MathSciNetCrossRefzbMATHGoogle Scholar
  37. Girault V, Scott LR (2003) A quasi-local interpolation operator preserving the discrete divergence. Calcolo 40:1–19MathSciNetCrossRefzbMATHGoogle Scholar
  38. Glowinski R, Le Tallec P (1989) Augmented Lagrangian and operator-splitting methods in nonlinear mechanics. SIAM studies in applied mathematics, vol 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, pp x+295Google Scholar
  39. Guzmán J, Neilan M (2014) Conforming and divergence-free Stokes elements on general triangular meshes. Math Comp 83:15–36MathSciNetCrossRefzbMATHGoogle Scholar
  40. Hansbo P, Szepessy A (1990) A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 84:175–192MathSciNetCrossRefzbMATHGoogle Scholar
  41. Heister T, Rapin G (2013) Efficient augmented Lagrangian-type preconditioning for the Oseen problem using grad-div stabilization. Int J Numer Methods Fluids 71:118–134MathSciNetCrossRefGoogle Scholar
  42. Hughes TJR, Franca LP (1987) 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 65:85–96MathSciNetCrossRefzbMATHGoogle Scholar
  43. 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 accommodating equal-order interpolations. Comput Methods Appl Mech Eng 59:85–99MathSciNetCrossRefzbMATHGoogle Scholar
  44. Hunter J (2007) Matplotlib: a 2D graphics environment. Comput Sci Eng 9:90–95CrossRefGoogle Scholar
  45. Jenkins EW, John V, Linke A, Rebholz LG (2014) On the parameter choice in grad-div stabilization for the Stokes equations. Adv Comput Math 40:491–516MathSciNetCrossRefzbMATHGoogle Scholar
  46. John V, Linke A, Merdon C, Neilan M, Rebholz LG (2016) On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Review (in press)Google Scholar
  47. Kellogg RB, Osborn JE (1976) A regularity result for the Stokes problem in a convex polygon. J Funct Anal 21:397–431MathSciNetCrossRefzbMATHGoogle Scholar
  48. Knobloch P (2001) Uniform validity of discrete Friedrichs’ inequality for general nonconforming finite element spaces. Numer Funct Anal Optim 22:107–126MathSciNetCrossRefzbMATHGoogle Scholar
  49. Král J, Wendland W (1986) Some examples concerning applicability of the Fredholm-Radon method in potential theory. Apl. Mat. 31:293–308MathSciNetzbMATHGoogle Scholar
  50. Lederer PL (2016) Pressure Robust Discretizations for Navier Stokes Equations: Divergence-free Reconstruction for Taylor-Hood Elements and High Order Hybrid Discontinuous Galerkin Methods. Diplomarbeit, Technische Universität WienGoogle Scholar
  51. Linke A (2014) On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput Methods Appl Mech Eng 268:782–800MathSciNetCrossRefzbMATHGoogle Scholar
  52. Linke A, Merdon C, Wollner W (2016a) Optimal L 2 velocity error estimate for a modified pressure-robust Crouzeix–Raviart Stokes element. IMA J Numer Anal (in press)Google Scholar
  53. Linke A, Matthies G, Tobiska L (2016b) Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors. ESAIM Math Model Numer Anal 50:289–309MathSciNetCrossRefzbMATHGoogle Scholar
  54. Neilan M (2015) Discrete and conforming smooth de Rham complexes in three dimensions. Math Comp 84:2059–2081MathSciNetCrossRefzbMATHGoogle Scholar
  55. Nitsche J (1968) Ein Kriterium für die Quasi-Optimalität des Ritzschen Verfahrens. Numer Math 11:346–348MathSciNetCrossRefzbMATHGoogle Scholar
  56. Olshanskii MA (2002) 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 191:5515–5536MathSciNetCrossRefzbMATHGoogle Scholar
  57. Olshanskii MA, Reusken A (2004) Grad-div stabilization for Stokes equations. Math Comp 73:1699–1718MathSciNetCrossRefzbMATHGoogle Scholar
  58. Olshanskii M, Lube G, Heister T, Löwe J (2009) Grad-div stabilization and subgrid pressure models for the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 198:3975–3988MathSciNetCrossRefzbMATHGoogle Scholar
  59. Qin J (1994) On the convergence of some low order mixed finite elements for incompressible fluids. PhD thesis, Department of Mathematics, Pennsylvania State UniversityGoogle Scholar
  60. Roos H-G, Stynes M, Tobiska L (2008) Robust numerical methods for singularly perturbed differential equations. Convection-diffusion-reaction and flow problems. Springer series in computational mathematics, vol 24, 2nd edn. Springer, Berlin, pp xiv+604Google Scholar
  61. Scott LR, Zhang S (1990) Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comp 54:483–493MathSciNetCrossRefzbMATHGoogle Scholar
  62. Tobiska L, Verfürth R (1996) Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations. SIAM J Numer Anal 33:107–127MathSciNetCrossRefzbMATHGoogle Scholar
  63. Vassilevski PS, Lazarov RD (1996) Preconditioning mixed finite element saddle-point elliptic problems. Numer Linear Algebra Appl 3:1–20MathSciNetCrossRefzbMATHGoogle Scholar
  64. Verchota GC, Vogel AL (2003) A multidirectional Dirichlet problem. J Geom Anal 13:495–520MathSciNetCrossRefzbMATHGoogle Scholar
  65. Verfürth R (1989) A posteriori error estimators for the Stokes equations. Numer Math 55:309–325MathSciNetCrossRefzbMATHGoogle Scholar
  66. Verfürth R (1994) A posteriori error estimation and adaptive mesh-refinement techniques. J Comput Appl Math 50:67–83MathSciNetCrossRefzbMATHGoogle Scholar
  67. Verfürth R (2013) A posteriori error estimation techniques for finite element methods. Numerical mathematics and scientific computation. Oxford University Press, Oxford, pp xx+393Google Scholar
  68. Zhang S (2005) A new family of stable mixed finite elements for the 3D Stokes equations. Math Comp 74:543–554MathSciNetCrossRefzbMATHGoogle Scholar
  69. Zhang S (2009) Bases for C0-P1 divergence-free elements and for C1-P2 finite elements on union jack grids. Accessed 20 July 2016

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Volker John
    • 1
    • 2
  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany

Personalised recommendations