Weak discrete maximum principle and \(L^\infty \) analysis of the DG method for the Poisson equation on a polygonal domain

  • Yuki ChibaEmail author
  • Norikazu Saito
Original Paper


We derive several \(L^\infty \) error estimates for the symmetric interior penalty discontinuous Galerkin method applied to the Poisson equation in a two-dimensional polygonal domain. Both local and global estimates are examined. The weak maximum principle (WMP) for the discrete harmonic function is also established. We prove our \(L^\infty \) estimates using this WMP and several \(W^{2,p}\) and \(W^{1,1}\) estimates for the Poisson equation. Numerical examples to validate our results are also presented.


Discontinuous Galerkin method Pointwise error estimate Maximum principle 

Mathematics Subject Classification

65N15 65N30 



We thank the anonymous reviewer for his/her valuable comments and suggestions to improve the quality of the paper. Yuki Chiba was supported by Program for Leading Graduate Schools, MEXT, Japan. Norikazu Saito was supported by JST CREST Grant number JPMJCR15D1, Japan, and JSPS KAKENHI Grant number 15H03635, Japan.


  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), vol. 140, second edn. Elsevier/Academic Press, Amsterdam (2003)Google Scholar
  2. 2.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/2002).
  3. 3.
    Ayuso, B., Marini, L.D.: Discontinuous Galerkin methods for advection–diffusion–reaction problems. SIAM J. Numer. Anal. 47(2), 1391–1420 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Badia, S., Hierro, A.: On discrete maximum principles for discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 286, 107–122 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Badia, S., Bonilla, J., Hierro, A.: Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes. Comput. Methods Appl. Mech. Eng. 320, 582–605 (2017). MathSciNetCrossRefGoogle Scholar
  6. 6.
    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).
  7. 7.
    Brézis, H., Strauss, W.A.: Semi-linear second-order elliptic equations in \(L^{1}\). J. Math. Soc. Jpn. 25, 565–590 (1973). CrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Z., Chen, H.: Pointwise error estimates of discontinuous Galerkin methods with penalty for second-order elliptic problems. SIAM J. Numer. Anal. 42(3), 1146–1166 (2004). MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. 4. North-Holland, Amsterdam (1978)Google Scholar
  10. 10.
    Ciarlet, P.G., Raviart, P.A.: Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2, 17–31 (1973). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grisvard, P.: Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain. Numerical Solution of P.D.E’s III, Proc. Third Sympos. (SYNSPADE), pp. 207–274 (1976)Google Scholar
  12. 12.
    Horváth, T.L., Mincsovics, M.E.: Discrete maximum principle for interior penalty discontinuous Galerkin methods. Cent. Eur. J. Math. 11(4), 664–679 (2013). MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kashiwabara, T., Kemmochi, T.: \(L^\infty \)- and \(W^{1,\infty }\)-error estimates of linear finite element method for Neumann boundary value problems in a smooth domain. ArXiv e-prints (2018)Google Scholar
  14. 14.
    Nitsche, J.A.: \(L_{\infty }\)-convergence of finite element approximation. In: Journées “Éléments Finis” (Rennes, 1975), p. 18. Univ. Rennes, Rennes (1975)Google Scholar
  15. 15.
    Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38(158), 437–445 (1982). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. In: Los Alamos Scientific Laboratory Report LA-UR-73-479 (1973).
  17. 17.
    Rivière, B.: Discontinuous Galerkin methods for solving elliptic and parabolic equations, Frontiers in Applied Mathematics, vol. 35. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008). Theory and implementation
  18. 18.
    Schatz, A.H.: A weak discrete maximum principle and stability of the finite element method in \(L_{\infty }\) on plane polygonal domains. I. Math. Comput. 34(149), 77–91 (1980). zbMATHGoogle Scholar
  19. 19.
    Schatz, A.H.: Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates. Math. Comput. 67(223), 877–899 (1998). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comput. 31(138), 414–442 (1977). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schechter, M.: On \(L^{p}\) estimates and regularity. I. Am. J. Math. 85, 1–13 (1963). CrossRefzbMATHGoogle Scholar
  22. 22.
    Scott, R.: Optimal \(L^{\infty }\) estimates for the finite element method on irregular meshes. Math. Comput. 30(136), 681–697 (1976). MathSciNetzbMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

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

Personalised recommendations