FETI Solvers for Non-standard Finite Element Equations Based on Boundary Integral Operators

  • Clemens Hofreither
  • Ulrich Langer
  • Clemens Pechstein
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 98)

Abstract

We present efficient Domain Decomposition solvers for a class of non-standard Finite Element Methods. These methods utilize PDE-harmonic trial functions in every element of a polyhedral mesh, and use boundary element techniques locally in order to generate the finite element stiffness matrices. For these reasons, the terms BEM-based FEM or Trefftz-FEM are sometimes used. In the present paper, we show that Finite Element Tearing and Interconnecting (FETI) methods can be used to solve the resulting linear systems in a quasi-optimal, robust and parallel manner. An important theoretical tool are spectral equivalences between FEM- and BEM-approximated Steklov–Poincaré operators.

Notes

Acknowledgement

The authors gratefully acknowledge the financial support by the Austrian Science Fund (FWF) under the grant DK W1214, project DK4.

References

  1. 1.
    Farhat, C., Roux, F.X.: A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32, 1205–1227 (1991)MATHCrossRefGoogle Scholar
  2. 2.
    Hofreither, C.: L 2 error estimates for a nonstandard finite element method on polyhedral meshes. J. Numer. Math. 19(1), 27–39 (2011)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Hofreither, C., Pechstein, C.: A rigorous error analysis of coupled FEM-BEM problems with arbitrary many subdomains. In: Apel, T., Steinbach, O. (eds.) Advanced Finite Element Methods and Applications, pp. 109–130. Springer, New York (2012)Google Scholar
  4. 4.
    Hofreither, C., Langer, U., Pechstein, C.: Analysis of a non-standard finite element method based on boundary integral operators. Electron. Trans. Numer. Anal. 37, 413–436 (2010)MATHMathSciNetGoogle Scholar
  5. 5.
    Klawonn, A., Widlund, O.B.: FETI and Neumann-Neumann iterative substructuring methods: connections and new results. Commun. Pure Appl. Math. 54(1), 57–90 (2001)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Langer, U., Steinbach, O.: Boundary element tearing and interconnecting method. Computing 71(3), 205–228 (2003)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Pechstein, C.: Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems. Springer, Heidelberg (2013)MATHCrossRefGoogle Scholar
  8. 8.
    Pechstein, C.: Shape-explicit constants for some boundary integral operators. Appl. Anal. 92(5), 949–974 (2013)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Rixen, D., Farhat, C.: A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems. Int. J. Numer. Methods Eng. 44(4), 489–516 (1999)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems – Finite and Boundary Elements. Springer, New York (2008)MATHCrossRefGoogle Scholar
  11. 11.
    Toselli, A., Widlund, O.: Domain Decoposition Methods – Algorithms and Theory. Springer, Berlin (2004)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Clemens Hofreither
    • 1
  • Ulrich Langer
    • 2
  • Clemens Pechstein
    • 2
  1. 1.Doctoral Program “Computational Mathematics” Johannes Kepler UniversityLinzAustria
  2. 2.Institute of Computational MathematicsJohannes Kepler UniversityLinzAustria

Personalised recommendations