Maximum Norm Estimates for Energy-Corrected Finite Element Method

  • Piotr SwierczynskiEmail author
  • Barbara Wohlmuth
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


Nonsmoothness of the boundary of polygonal domains limits the regularity of the solutions of elliptic problems. This leads to the presence of the so-called pollution effect in the finite element approximation, which results in a reduced convergence order of the scheme measured in the L2 and L-norms, compared to the best-approximation order. We show that the energy-correction method, which is known to eliminate the pollution effect in the L2-norm, yields the same convergence order of the finite element error as the best approximation also in the L-norm. We confirm the theoretical results with numerical experiments.



We gratefully acknowledge the support of the German Research Foundation (DFG) through the grant WO 671/11-1 and, together with the Austrian Science Fund, through the IGDK1754 Training Group. We would also like to thank Dr Johannes Pfefferer for many fruitful and helpful discussions.


  1. 1.
    T. Apel, J. Pfefferer, A. Rösch, Finite element error estimates on the boundary with application to optimal control. Math. Comput. 84(291), 33–70 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    H. Blum, M. Dobrowolski, On finite element methods for elliptic equations on domains with corners. Computing 28(1), 53–63 (1982)MathSciNetCrossRefGoogle Scholar
  3. 3.
    P.G. Ciarlet, J.L. Lions, Finite Element Methods (Part 1). Handbook of Numerical Analysis, vol. II (North Holland, Amsterdam, 1991)Google Scholar
  4. 4.
    H. Egger, U. Rüde, B. Wohlmuth, Energy-corrected finite element methods for corner singularities. SIAM J. Numer. Anal. 52(1), 171–193 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    T. Horger, P. Pustejovska, B.Wohlmuth, Higher order energy-corrected finite element methods. ArXiv e-prints (2017).
  6. 6.
    V.A. Kondratiev, Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 16, 227–313 (1967)MathSciNetGoogle Scholar
  7. 7.
    V. Kozlov, V.G. Maz’ya, J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs, vol. 85 (American Mathematical Society, Providence, 2001)Google Scholar
  8. 8.
    J. Nitsche, A.H. Schatz, Interior estimates for Ritz-Galerkin methods. Math. Comput. 28, 973–958 (1974)MathSciNetCrossRefGoogle Scholar
  9. 9.
    U. Rüde, C. Waluga, B. Wohlmuth, Nested Newton strategies for energy-corrected finite element methods. SIAM J. Sci. Comput. 36(4), A1359–A1383 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A.H. Schatz, L.B. Wahlbin, Interior maximum norm estimates for finite element methods. Math. Comput. 31(138), 414–442 (1977)MathSciNetCrossRefGoogle Scholar
  11. 11.
    A.H. Schatz, L.B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. Part 1. Math. Comput. 32(141), 73–109 (1978)Google Scholar
  12. 12.
    A.H. Schatz, L.B. Wahlbin, Maximum norm estimates in the finite element method on plane polygonal domains. Part 2, Refinements. Math. Comput. 33(146), 465–492 (1979)Google Scholar
  13. 13.
    C. Zenger, H. Gietl, Improved difference schemes for the Dirichlet problem of Poisson’s equation in the neighbourhood of corners. Numer. Math. 30(3), 315–332 (1978)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute for Numerical MathematicsTechnical University of MunichMünchenGermany

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