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Weighted FEM for Two-Dimensional Elasticity Problem with Corner Singularity

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Numerical Mathematics and Advanced Applications ENUMATH 2015

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 112))

Abstract

In this paper we consider homogeneous Dirichlet problem for the Lamé system with singularity caused by the reentrant corner to the boundary of the two-dimensional domain. For this problem we define the solution as a R ν -generalized one; we state its existence and uniqueness in the weighted set \(\mathring{\mathbf{W}}_{2,\nu }^{1}(\varOmega,\delta )\). On the basis of the R ν -generalized solution we construct weighted finite element method. We prove that the approximate solution converges to the exact one with the rate O(h) in the norm of W 2, ν 1(Ω), and results of numerical experiments are presented.

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References

  1. T. Apel, A.-M. Sändig, J.R. Whiteman, Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci. 19 (1), 63–85 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. A.Y. Bespalov, V.A. Rukavishnikov, The exponential rate of convergence of the finite-element method for the Dirichlet problem with singularity of the solution. Dokl. Math. 62 (2), 266–270 (2000)

    MathSciNet  MATH  Google Scholar 

  3. J.H. Bramble, M. Zlámal, Triangular elements in the finite element method. Math. Comput. 24 (112), 809–820 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Morin, R.H. Hochetto, K.G. Siebert, Convergence of adaptive finite element methods. SIAM Rev. 44 (5), 631–658 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Nguyen-Xuan, R.G. Liu, S. Bordas, S. Natarajan, T. Rabczuk, An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order. Comput. Methods Appl. Mech. Eng. 253, 252–273 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Rössle, Corner singularities and regulatiy of weak solutions for the two-dimensional Lamé equations on domains with angular corners. J. Elast. 60 (1), 57–75 (2000)

    Article  MATH  Google Scholar 

  7. V.A. Rukavishnikov, On the differential properties of R ν -generalized solution of Dirichlet problem. Dokl. Akad. Nauk SSSR 309 (6), 1318–1320 (1989)

    MathSciNet  Google Scholar 

  8. V.A. Rukavishnikov, The Dirichlet problem for a second-order elliptic equation with noncoordinated degeneration of the input data. Differ. Equ. 32 (3), 406–412 (1996)

    MathSciNet  MATH  Google Scholar 

  9. V.A. Rukavishnikov, On the existence and uniqueness of an R ν -generalized solution of a boundary value problem with uncoordinated degeneration of the input data. Dokl. Math. 90 (2), 562–564 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. V.A. Rukavishnikov, E.V. Kuznetsova, Coercive estimate for a boundary value problem with noncoordinated degeneration of the data. Differ. Equ. 43 (4), 550–560 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. V.A. Rukavishnikov, E.V. Kuznetsova, The R ν -generalized solution of a boundary value problem with a singularity belongs to the space W 2, ν+β∕2+k+1 k+2(Ω, δ). Differ. Equ. 45 (6), 913–917 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. V.A. Rukavishnikov, A.O. Mosolapov, New numerical method for solving time-harmonic Maxwell equations with strong singularity. J. Comput. Phys. 231 (6), 2438–2448 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. V.A. Rukavishnikov, A.O. Mosolapov, Weighted edge finite element method for Maxwell’s equations with strong singularity. Dokl. Math. 87 (2), 156–159 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. V.A. Rukavishnikov, S.G. Nikolaev, Weighted finite element method for an elasticity problem with singularity. Dokl. Math. 88 (3), 705–709 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. V.A. Rukavishnikov, S.G. Nikolaev, On the R ν -generalized solution of the Lamé system with corner singularity. Dokl. Math. 92 (1), 421–423 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. V.A. Rukavishnikov, H.I. Rukavishnikova, Finite-element method for the 1st boundary-value problem with the coordinated degeneration of the initial data. Dokl. Akad. Nauk 338 (6), 731–733 (1994)

    MATH  Google Scholar 

  17. V.A. Rukavishnikov, H.I. Rukavishnikova, The finite element method for a boundary value problem with strong singularity. J. Comput. Appl. Math. 234 (9), 2870–2882 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. V.A. Rukavishnikov, H.I. Rukavishnikova, On the error estimation of the finite element method for the boundary value problems with singularity in the Lebesgue weighted space. Numer. Funct. Anal. Optim. 34 (12), 1328–1347 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Szabó, I. Babuška, Finite Element Analysis (Wiley, New York, 1991)

    MATH  Google Scholar 

  20. Z. Yosibash, Singularities in Elliptic Boundary Value Problems and Elasticity and Their Connection with Failure Initiation. Intersisciplinary Applied Mathematics, vol. 37 (Springer, New York, 2012)

    Google Scholar 

  21. O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, 6th edn. (Elsevier, Oxford/Boston, 2005)

    MATH  Google Scholar 

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Acknowledgements

This work was made under the frame of project 16-11-10008 of the RSCF.

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Authors and Affiliations

  1. Russian Academy of Sciences, Kim U Chen str. 65, Khabarovsk, 680000, Russia

    Viktor A. Rukavishnikov

  2. Far Eastern State Transport University, Khabarovsk, Russia

    Viktor A. Rukavishnikov

Authors
  1. Viktor A. Rukavishnikov
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Corresponding author

Correspondence to Viktor A. Rukavishnikov .

Editor information

Editors and Affiliations

  1. Mathematics & Applied Mathematics, Middle East Technical University Mathematics & Applied Mathematics, Ankara, Turkey

    Bülent Karasözen

  2. Dept of Computer Engg, Middle East Technical Univ Dept of Computer Engg, Cankaya, Ankara, Turkey

    Murat Manguoğlu

  3. Middle East Technical University Mathematics & Institute of Applied Mathe, Ankara, Turkey

    Münevver Tezer-Sezgin

  4. Civil Engineering & Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Serdar Göktepe

  5. Institute of Applied Mathematics, Middle East Technical University Institute of Applied Mathematics, Ankara, Turkey

    Ömür Uğur

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Rukavishnikov, V.A. (2016). Weighted FEM for Two-Dimensional Elasticity Problem with Corner Singularity. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_39

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