The inf-sup condition and error estimates of the Nitsche method for evolutionary diffusion–advection-reaction equations

  • Yuki UedaEmail author
  • Norikazu Saito
Original Paper Area 2


The Nitsche method is a method of “weak imposition” of the inhomogeneous Dirichlet boundary conditions for partial differential equations. This paper explains stability and convergence study of the Nitsche method applied to evolutionary diffusion–advection-reaction equations. We mainly discuss a general space semidiscrete scheme including not only the standard finite element method but also Isogeometric Analysis. Our method of analysis is a variational one that is a popular method for studying elliptic problems. The variational method enables us to obtain the best approximation property directly. Actually, results show that the scheme satisfies the inf-sup condition and Galerkin orthogonality. Consequently, the optimal order error estimates in some appropriate norms are proven under some regularity assumptions on the exact solution. We also consider a fully discretized scheme using the backward Euler method. Numerical example demonstrate the validity of those theoretical results.


Diffusion–advection-reaction equation Inf-sup condition IGA 

Mathematics Subject Classification

65M12 65M60 



We thank the anonymous reviewer for his/her valuable comments and suggestions to improve the quality of the paper. This study was supported by JST CREST Grant Number JPMJCR15D1 and JSPS KAKENHI Grant Number 15H03635. The first author was also supported by the Program for Leading Graduate Schools, MEXT, Japan.


  1. 1.
    Bazilevs, Y., Hughes, T.J.R.: Weak imposition of Dirichlet boundary conditions in fluid mechanics. Comput. Fluids 36(1), 12–26 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bazilevs, Y., Michler, C., Calo, V.M., Hughes, T.J.R.: Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Comput. Methods Appl. Mech. Eng. 196(49–52), 4853–4862 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beirão da Veiga, L., Buffa, A., Sangalli, G., Vázquez, R.: Mathematical analysis of variational isogeometric methods. Acta Numer. 23, 157–287 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)CrossRefGoogle Scholar
  6. 6.
    Burman, E.: A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions. SIAM J. Numer. Anal. 50(4), 1959–1981 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Burman, E., Fernández, M.A., Hansbo, P.: Continuous interior penalty finite element method for Oseen’s equations. SIAM J. Numer. Anal. 44(3), 1248–1274 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Choudury, G., Lasiecka, I.: Optimal convergence rates for semidiscrete approximations of parabolic problems with nonsmooth boundary data. Numer. Funct. Anal. Optim. 12(5–6), 469–485 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, New York (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5. Springer, Berlin (1992). Evolution problems. I, With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Translated from the French by Alan CraigGoogle Scholar
  12. 12.
    Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer, New York (2004)CrossRefzbMATHGoogle Scholar
  13. 13.
    Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)Google Scholar
  14. 14.
    Evans, J.A., Hughes, T.J.R.: Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements. Numer. Math. 123(2), 259–290 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Freund, J., Stenberg, R.: On weakly imposed boundary conditions for second order problems. In: Cecchi, M., et al. (eds.) Proceedings of the Ninth International Conference on Finite Elements in Fluids, pp. 327–336. Università di Padova, Padova (1995)Google Scholar
  16. 16.
    Heinrich, B., Jung, B.: Nitsche mortaring for parabolic initial-boundary value problems. Electron. Trans. Numer. Anal. 32, 190–209 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilr äumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971). Collection of articles dedicated to Lothar Collatz on his sixtieth birthdayGoogle Scholar
  18. 18.
    Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous $hp$ finite element method for diffusion problems. J. Comput. Phys. 146(2), 491–519 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Saito, N.: Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations. arXiv:1710.10543 Google Scholar
  20. 20.
    Schumaker, L.L.: Spline Functions: Basic Theory. Cambridge Mathematical Library, 3rd edn. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  21. 21.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  22. 22.
    Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987). Translated from the German by C. B. Thomas and M. J. ThomasGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

Personalised recommendations