A Non-standard Finite Element Method Based on Boundary Integral Operators

  • Clemens Hofreither
  • Ulrich Langer
  • Clemens Pechstein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7116)

Abstract

This paper provides an overview over our results on the construction and analysis of a non-standard finite element method that is based on the use of boundary integral operators for constructing the element stiffness matrices. This approach permits polyhedral element shapes as well as meshes with hanging nodes. We consider the diffusion equation and convection-diffusion-reaction problems as our model problems, but the method can also be generalized to more general problems like systems of partial differential equations. We provide a rigorous H1- and L2-error analysis of the method for smooth and non-smooth solutions. This a priori discretization error analysis is only done for the diffusion equation. However, our numerical results also show good performance of our method for convection-dominated diffusion problems.

Keywords

non-standard FEM boundary integral operators Trefftz method polyhedral meshes convection-diffusion-reaction problems 

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References

  1. 1.
    Bebendorf, M.: A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen 22(4), 751–756 (2003)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ciarlet, P.G.: The finite element method for elliptic problems. Studies in Mathematics and its Applications, vol. 4. North-Holland, Amsterdam (1987)Google Scholar
  3. 3.
    Copeland, D.M.: Boundary-element-based finite element methods for Helmholtz and Maxwell equations on general polyhedral meshes. Int. J. Appl. Math. Comput. Sci. 5(1), 60–73 (2009)MathSciNetGoogle Scholar
  4. 4.
    Copeland, D.M., Langer, U., Pusch, D.: From the Boundary Element Method to Local Trefftz Finite Element Methods on Polyhedral Meshes. In: Bercovier, M., Gander, M.J., Kornhuber, R., Widlund, O. (eds.) Domain Decomposition Methods in Science and Engineering XVIII. Lecture Notes in Computational Science and Engineering, vol. 70, pp. 315–322. Springer, Heidelberg (2009)Google Scholar
  5. 5.
    Costabel, M.: Symmetric Methods for the Coupling of Finite Elements and Boundary Elements. In: Brebbia, C., Wendland, W., Kuhn, G. (eds.) Boundary Elements IX, pp. 411–420. Springer, Heidelberg (1987)Google Scholar
  6. 6.
    Girault, V., Raviart, P.: Finite element methods for Navier-Stokes equations. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)MATHCrossRefGoogle Scholar
  7. 7.
    Gordon, D., Gordon, R.: Row scaling as a preconditioner for some nonsymmetric linear systems with discontinuous coefficients. J. Computational Applied Mathematics 234(12), 3480–3495 (2010)MATHCrossRefGoogle Scholar
  8. 8.
    Hofreither, C.: L 2 error estimates for a nonstandard finite element method on polyhedral meshes. J. Numer. Math. 19(1), 27–39 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hofreither, C., Langer, U., Pechstein, C.: Analysis of a non-standard finite element method based on boundary integral operators. Electronic Transactions on Numerical Analysis 37, 413–436 (2010), http://etna.mcs.kent.edu/vol.37.2010/pp413-436.dir MathSciNetMATHGoogle Scholar
  10. 10.
    Hofreither, C., Langer, U., Pechstein, C.: A non-standard finite element method for convection-diffusion-reaction problems on polyhedral meshes. In: Proceedings of the Third Conference of the Euro-American Consortium for Promoting the Application of Mathematics in Technical and Natural Sciences. American Institute of Physics (2011)Google Scholar
  11. 11.
    Hsiao, G.C., Steinbach, O., Wendland, W.L.: Domain decomposition methods via boundary integral equations. Journal of Computational and Applied Mathematics 125(1-2), 521–537 (2000)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hsiao, G.C., Wendland, W.L.: Domain decomposition in boundary element methods. In: Glowinski, R., Kuznetsov, Y.A., Meurant, G., Périaux, J., Widlund, O.B. (eds.) Proceedings of the Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, Moscow, May 21-25, 1990, pp. 41–49. SIAM, Philadelphia (1991)Google Scholar
  13. 13.
    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer, Heidelberg (2008)MATHCrossRefGoogle Scholar
  14. 14.
    Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147, 71–88 (1981)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  16. 16.
    Pechstein, C.: Shape-explicit constants for some boundary integral operators. Appl. Anal. (December 2011) (published online), doi:10.1080/00036811.2011.643781Google Scholar
  17. 17.
    Sauter, S.A., Schwab, C.: Boundary Element Methods. Springer Series in Computational Mathematics, vol. 39. Springer, Heidelberg (2011)MATHCrossRefGoogle Scholar
  18. 18.
    Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements. Springer, New York (2008)MATHCrossRefGoogle Scholar
  19. 19.
    Thrun, A.: Über die Konstanten in Poincaréschen Ungleichungen. Master’s thesis, Ruhr-Universität Bochum, Bochum (2003), http://www.ruhr-uni-bochum.de/num1/files/theses/da_thrun.pdf
  20. 20.
    Trefftz, E.: Ein Gegenstück zum Ritzschen Verfahren. In: Verh. d. 2. Intern. Kongr. f. Techn. Mech., Zürich, pp. 131–137 (1926)Google Scholar
  21. 21.
    Veeser, A., Verfürth, R.: Poincaré constants of finite element stars. IMA J. Numer. Anal. (2011), first published online May 30, doi:10.1093/imanum/drr011Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Clemens Hofreither
    • 1
  • Ulrich Langer
    • 2
  • Clemens Pechstein
    • 2
  1. 1.DK Computational MathematicsJohannes Kepler University LinzLinzAustria
  2. 2.Institute of Computational MathematicsJohannes Kepler University LinzLinzAustria

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