Skip to main content

A constructive a priori error estimation for finite element discretizations in a non-convex domain using singular functions

Abstract

In solving elliptic problems by the finite element method in a bounded domain which has a re-entrant corner, the rate of convergence can be improved by adding a singular function to the usual interpolating basis. When the domain is enclosed by line segments which form a corner of π/2 or 3π/2, we have obtained an explicit a prioriH 10 error estimation ofO(h) and anL 2 error estimation ofO(h 2) for such a finite element solution of the Poisson equation. Particularly, we emphasize that all constants in our error estimates are numerically determined, which plays an essential role in the numerical verification of solutions to non-linear elliptic problems.

This is a preview of subscription content, access via your institution.

References

  1. R.A. Adams, Sobolev Spaces. Academic Press, New York, 1975.

    MATH  Google Scholar 

  2. I. Babuska, R.B. Kellog and J. Pitkaranta, Direct and inverse error estimates for finite elements with mesh refinement. Numer. Math.,33 (1979), 447–471.

    MATH  Article  MathSciNet  Google Scholar 

  3. H. Blum and M. Dobrowolski, On finite element methods for elliptic equations on domains with corners. Computing,28 (1982), 53–63.

    MATH  Article  MathSciNet  Google Scholar 

  4. S.C. Brenner, Multigrid methods for the computation of singular solutions and stress intensity factors, I: Corner singularities. Math. Comp.,68 (1999), 559–583.

    MATH  Article  MathSciNet  Google Scholar 

  5. Z. Cai and S. Kim, A finite element method using singular function for the Poisson equation: corner singularities. SIAM J. Numer. Anal.,39 (2001), 286–299.

    MATH  Article  MathSciNet  Google Scholar 

  6. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978.

    MATH  Google Scholar 

  7. M. Dauge, S. Nicaise, M. Bourland and M.S. Lubuma, Coefficients of the singularities for elliptic boundary value problems on domains with conical points, III: Finite element methods on polygonal domains. SIAM J. Numer. Anal.,29 (1992), 136–155.

    MATH  Article  MathSciNet  Google Scholar 

  8. G. Fix, S. Gulati and G.I. Wakoff, On the use of singular functions with the finite element method. J. Comp. Phys.,13 (1973), 209–228.

    Article  MathSciNet  Google Scholar 

  9. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Publishing, Boston, 1985.

    MATH  Google Scholar 

  10. P. Grisvard, Singularities in Boundary Value Problems. RMA,22, Masson, Paris, 1992.

    MATH  Google Scholar 

  11. O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, 1969.

  12. M.T. Nakao, A numerical verification method for the existence of weak solutions for non-linear boundary value problems. J. Math. Anal. Appl.,164 (1992), 489–507.

    MATH  Article  MathSciNet  Google Scholar 

  13. M. Schultz, Spline Analysis. Prentice-Hall, 1973.

  14. M. Tabata and M. Yamaguti, Approximate solution of the second order elliptic differential equation in a domain with piecewise smooth boundary by the finite-element method using singularity functions. Theor. and Appl. Mech.,22 (1974), 165–173.

    MathSciNet  Google Scholar 

  15. H. Takahasi and M. Mori, Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Soc.,9 (1974), 721–741.

    Article  MathSciNet  Google Scholar 

  16. N. Yamamoto and M.T. Nakao, Numerical verifications of solutions for elliptic equations in nonconvex polygonal domains. Numer. Math.,65 (1993), 503–521.

    MATH  Article  MathSciNet  Google Scholar 

  17. Y. Watanabe and M.T. Nakao, Numerical verifications of solutions for nonlinear elliptic equations. Japan J. Indust. Appl. Math.,10 (1993), 165–178.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kenta Kobayashi.

Additional information

This research is partly supported by the 21st Century COE program of Faculty of Mathematics, Kyushu University and Grant-in-Aid for Young Scientists (B), 19740052 by the Ministry of Education, Science, Sports and Culture.

About this article

Cite this article

Kobayashi, K. A constructive a priori error estimation for finite element discretizations in a non-convex domain using singular functions. Japan J. Indust. Appl. Math. 26, 493–516 (2009). https://doi.org/10.1007/BF03186546

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03186546

Key words

  • finite element method
  • a priori error estimation
  • Poisson equation