Skip to main content
SpringerLink
Log in

The Approximation and Computation of a Basis of the Trace Space H 1/2

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

We present a method for construction of an approximate basis of the trace space H 1/2 based on a combination of the Steklov spectral method and a finite element approximation. Specifically, we approximate the Steklov eigenfunctions with respect to a particular finite element basis. Then solutions of elliptic boundary value problems with Dirichlet boundary conditions can be efficiently and accurately expanded in the discrete Steklov basis. We provide a reformulation of the discrete Steklov eigenproblem as a generalized eigenproblem that we solve by the implicitly restarted Arnoldi method of ARPACK. We include examples highlighting the computational properties of the proposed method for the solution of elliptic problems on bounded domains using both a conforming bilinear finite element and a non-conforming harmonic finite element. In addition, we document the efficiency of the proposed method by solving a Dirichlet problem for the Laplace equation on a densely perforated domain.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auchmuty G. (2004). Steklov Eigenproblems, and representation of solutions of elliptic, boundary value problems. Num. Func. Anal. Opt. 25(3-4):321–348

    Article  MATH  MathSciNet  Google Scholar 

  2. Auchmuty, G., (2005). Spectral characterization of the trace, spaces H s(∂Ω), SIAM J. Math. Anal. (in Press).

  3. Auchmuty, G., and Klouček, P. (2006). Generalized harmonic, functions, and the dewetting of thin films. App Math. Opt. (in Press).

  4. Buttazzo, G. On the existence of minimizing domains for some shape optimization problems, ESAIM: Proceedings Actes du 29éme Congrés d’Analyse Numérique 3 (1998), 51–64.

  5. Cao L.-Q., Cui J.-Z. (2004). Assymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for the second order elliptic equations in perforated domains. Numer. Math. 96(3):525–581

    Article  MATH  MathSciNet  Google Scholar 

  6. Carstensen, C., and Sauter, S. A. (2001). A posteriori error analysis for elliptic PDEs on domains with complicated structures. Math. Comp. to appear.

  7. Cioranescu, D., and Murat, F. (1997). Topics in the mathematical modelling of composite materials, A. Cherkaev (ed.), Prog. Nonlinear Differ. Equ. Appl., ch. A strange term coming from nowhere, pp. 45–93, Birkhaüser.

  8. Ericsson T., Ruhe A. (1980). The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. of Comp. 35(152):1251–1268

    Article  MATH  MathSciNet  Google Scholar 

  9. Groemer H. (1996). Geometric applications of Fourier series and spherical harmonics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  10. Jikov V.V., Kozlov S.M., Olejnik O.A. (1994). Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin

    Google Scholar 

  11. Klouček P., Li B., Luskin M. (1996). Analysis of a class of nonconforming finite elements for crystalline microstructures. Math. Comp. 65(215):1111–1135

    Article  MathSciNet  Google Scholar 

  12. Knyazev, A., and Windlund, O. (2001). Lavrentiev regularization + Ritz approximation = Uniform finite element error estimates for differential equations with rough coefficients. Math. Comp.

  13. Lehoucq, R. B., Sorensen, D. C., and Yang, C. (1998). ARPACK users’ guide: Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods, SIAM, Philadelphia.

  14. Magenes, E., and Lions, J. L. (1968). Problèmes aux limites non homogènes et applications, Dunod, Paris.

  15. Piat V.Ch., Codegone M. (2003). Scattering problem in a domain with small holes. Rev. R. Acad. Cien. Serie A., Mat. 97(3):447–454

    MATH  Google Scholar 

  16. Sorensen D.C. (1992). Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix. Anal. Appl. 13:357–385

    Article  MATH  MathSciNet  Google Scholar 

  17. Strouboulis, T., Zhang, L., and Babuška, I. (2004). p-version of the generalized FEM using mesh-based handbooks with applications to multiscale problems.

  18. Wendland, W. L. (1999). Mathematical Aspects of Boundary Element Methods, CRC Research Notes in Mathematics (Paperback), Chapman & Hall.

Download references

Author information

Authors and Affiliations

  1. Texas Learning and Computational Center, University of Houston, 4800, Calhoun, TX, 77204, USA

    Petr Klouček

  2. Institut de Mathèmatiques, Universitè de Neuchâtel, Rue Emile Argand 11, CH-2007, Neuchâtel, Switzerland

    Petr Klouček

  3. Department of Computational and Applied Math, Rice University, 6100 Main Street, Houston, TX, 77005, USA

    Danny C. Sorensen

  4. Department of Math and Statistics, Coastal Carolina University, P.O. Box 261954, Conway, SC, 29528, USA

    Jennifer L. Wightman

Authors
  1. Petr Klouček
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Danny C. Sorensen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jennifer L. Wightman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Petr Klouček.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Klouček, P., Sorensen, D.C. & Wightman, J.L. The Approximation and Computation of a Basis of the Trace Space H 1/2 . J Sci Comput 32, 73–108 (2007). https://doi.org/10.1007/s10915-006-9118-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-006-9118-4

Keywords

Search

Navigation