Skip to main content
SpringerLink
Log in

The Finite Element Method for Boundary Value Problems with Strong Singularity and Double Singularity

  • Conference paper
Book cover Numerical Analysis and Its Applications (NAA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8236))

Included in the following conference series:

Abstract

A boundary value problem is said to possess strong singularity if its solution u does not belong to the Sobolev space \(W^1_2\) (H 1) or, in other words, the Dirichlet integral of the solution u diverges.

We consider the boundary value problems with strong singularity and with double singularity caused the discontinuity of coefficients in the equation on the domain with slot and presence of the corners equal 2π on boundary of this domain.

The schemes of the finite element method is constructed on the basis of the definition on R ν -generalized solution to these problems, and the finite element space contains singular power functions. The rate of convergence of the approximate solution to the R ν -generalized solution in the norm of the Sobolev weighted space is established and, finally, results of numerical experiments are presented.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Assous, F., Ciarlet Jr., P., Segré, J.: Numerical solution to the time-dependent Maxwell equations in two-dimentional singular domains: the singular complement method. J. Comp. Phys. 161, 218–249 (2000)

    Article  MATH  Google Scholar 

  2. Costabel, M., Dauge, M., Schwab, C.: Exponential convergence of hp-FEM for Maxwell equations with weighted regularization in polygonal domains. Math. Models Meth. Appl. Sci. 15, 575–622 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arroyo, D., Bespalov, A., Heuer, N.: On the finite element method for elliptic problems with degenerate and singular coefficients. Math. Comp. 76, 509–537 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Li, H., Nistor, V.: Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM. J. Comp. Appl. Math. 224, 320–338 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Rukavishnikov, V.A., Mosolapov, A.O.: New numerical method for solving time-harmonic Maxwell equations with strong singularity. Journal of Computational Physics 231, 2438–2448 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Rukavishnikov, V.A.: On a weighted estimate of the rate of convergence of difference schemes. Sov. Math. Dokl. 22, 826–829 (1986)

    Google Scholar 

  7. Rukavishnikov, V.A.: On differentiability properties of an R ν -generalized solution of Dirichlet problem. Sov. Math. Dokl. 40, 653–655 (1990)

    MathSciNet  MATH  Google Scholar 

  8. Rukavishnikov, V.A.: On the Dirichlet problem for the second order elliptic equation with noncoordinated degeneration of input data. Differ. Equ. 32, 406–412 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Rukavishnikov, V.A.: On the uniqueness of R ν -generalized solution for boundary value problem with non-coordinated degeneration of the input data. Dokl. Math 63, 68–70 (2001)

    Google Scholar 

  10. Rukavishnikov, V.A., Ereklintsev, A.G.: On the coercivity of the R ν -generalized solution of the first boundary value problem with coordinated degeneration of the input data. Differ. Equ. 41, 1757–1767 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rukavishnikov, V.A., Kuznetsova, E.V.: Coercive estimate for a boundary value problem with noncoordinated degeneration of the data. Differ. Equ. 43, 550–560 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rukavishnikov, V.A., Kuznetsova, E.V.: The R ν -generalized solution of a boundary value problem with a singularity belongs to the space \({W}^{k+2}_{2,\nu+\beta/2+k+1}(\Omega,\delta)\). Differ. Equ. 45, 913–917 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rukavishnikov, V.A., Rukavishnikova, E.I.: The finite element method for the first boundary value problem with compatible degeneracy of the input data. Russ. Acad. Sci., Dokl. Math. 50, 335–339 (1995)

    MathSciNet  Google Scholar 

  14. Rukavishnikov, V.A., Kuznetsova, E.V.: A finite element method scheme for boundary value problems with noncoordinated degeneration of input data. Numer. Anal. Appl. 2, 250–259 (2009)

    Article  Google Scholar 

  15. Rukavishnikov, V.A., Rukavishnikova, H.I.: The finite element method for a boundary value problem with strong singularity. J. Comp. Appl. Math. 234, 2870–2882 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rukavishnikov, V.A.: On differential properties R ν -generalized solution of the Dirichlet problem with coordinated degeneration of the input data. ISRN Mathematical Analysis, Article ID 243724, 18 p. (2011), doi: 10.5402/2011/243724

    Google Scholar 

  17. Rukavishnikov, V.A., Rukavishnikova, H.I.: The numerical method for boundary value problem with double singularity. Inform. and Control Systems 17, 47–52 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computing Center, Far-Eastern Branch, Russian Academy of Sciences, Kim-U-Chena st. 65, 680000, Khabarovsk, Russian Federation

    Viktor A. Rukavishnikov & Elena I. Rukavishnikova

  2. Far Eastern State Transport University, ul. Serysheva 47, Khabarovsk, 680021, Russia

    Viktor A. Rukavishnikov & Elena I. Rukavishnikova

Authors
  1. Viktor A. Rukavishnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elena I. Rukavishnikova
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technlogies, Bulgarian Academy of Sciences, Acad. G. Bonchev, Bl25A, 1113, Sofia, Bulgaria

    Ivan Dimov

  2. Department of Applied Analysis, Pázm\’{a}ny P\’{e}ter s\’{e}t\’{a}ny 1/C,, Eötvös Loránd University, 1117, Budapest, Hungary

    István Faragó

  3. University of Rousse, 8 Studentska Str.,, 7017, Rousse, Bulgaria

    Lubin Vulkov

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Rukavishnikov, V.A., Rukavishnikova, E.I. (2013). The Finite Element Method for Boundary Value Problems with Strong Singularity and Double Singularity. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2012. Lecture Notes in Computer Science, vol 8236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41515-9_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-41515-9_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-41514-2

  • Online ISBN: 978-3-642-41515-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation