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

  • Viktor A. Rukavishnikov
  • Elena I. Rukavishnikova
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)

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.

Keywords

Finite element method the Rν-generalized solution singularity of solution 

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References

  1. 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)CrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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. 7.
    Rukavishnikov, V.A.: On differentiability properties of an R ν-generalized solution of Dirichlet problem. Sov. Math. Dokl. 40, 653–655 (1990)MathSciNetzbMATHGoogle Scholar
  8. 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)MathSciNetzbMATHGoogle Scholar
  9. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)MathSciNetGoogle Scholar
  14. 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)CrossRefGoogle Scholar
  15. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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: 10.5402/2011/243724Google Scholar
  17. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Viktor A. Rukavishnikov
    • 1
    • 2
  • Elena I. Rukavishnikova
    • 1
    • 2
  1. 1.Computing Center, Far-Eastern BranchRussian Academy of SciencesKhabarovskRussian Federation
  2. 2.Far Eastern State Transport UniversityKhabarovskRussia

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