Advertisement

On Robust Error Estimation for Singularly Perturbed Fourth-Order Problems

  • Sebastian Franz
  • Hans-Görg Roos
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 120)

Abstract

Recently, several classes of fourth order singularly perturbed problems were considered and uniform convergence in the associated energy norm as well as in a balanced norm was proved. In this proceedings paper we will extend some results by looking into L -bounds and postprocessing.

References

  1. 1.
    Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2(4), 556–581 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Franz, S.: Superconvergence using pointwise interpolation in convection-diffusion problems. Appl. Numer. Math. 76, 132–144 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Franz, S., Roos, H.-G.: Robust error estimation in energy and balanced norms for singularly perturbed fourth order problems. Comput. Math. Appl. 72, 233–247 (2016)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Franz, S., Roos, H.-G., Wachtel, A.: A C 0 interior penalty method for a singularly-perturbed fourth-order elliptic problem on a layer-adapted mesh. Numer. Methods Partial Differ. Equ. 30(3), 838–861 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Kopteva, N.: The two-dimensional Sobolev inequality in the case of an arbitrary grid. Zh. Vychisl. Mat. Mat. Fiz. 38(4), 596–599 (1998)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Lin, Q., Yan, N., Zhou, A.: A rectangle test for interpolated element analysis. In: Proc. Syst. Sci. Eng., pp. 217–229. Great Wall (H.K.) Culture Publish Co. (1991)Google Scholar
  7. 7.
    Roos, H.-G., Linß, T.: Sufficient conditions for uniform convergence on layer-adapted grids. Computing 63, 27–45 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Tobiska, L.: Analysis of a new stabilized higher order finite element method for advection-diffusion equations. Comput. Methods Appl. Mech. Eng. 196, 538–550 (2006)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut Numerical MathematicsTechnische Universität DresdenDresdenGermany
  2. 2.Institut für Mathematik, BTU CottbusCottbusGermany

Personalised recommendations