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The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains

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Abstract

This paper is the second part of a threefold article, aimed at solving numerically the Poisson problem in three-dimensional prismatic or axisymmetric domains. In the first part of this series, the Fourier Singular Complement Method was introduced and analysed, in prismatic domains. In this second part, the FSCM is studied in axisymmetric domains with conical vertices, whereas, in the third part, implementation issues, numerical tests and comparisons with other methods are carried out. The method is based on a Fourier expansion in the direction parallel to the reentrant edges of the domain, and on an improved variant of the Singular Complement Method in the 2D section perpendicular to those edges. Neither refinements near the reentrant edges or vertices of the domain, nor cut-off functions are required in the computations to achieve an optimal convergence order in terms of the mesh size and the number of Fourier modes used.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. Dover, New York (1965)

  2. Assous, F., Degond, P., Heintzé, E., Raviart, P.-A., Segré, J.:On a finite-element method for solving the three-dimensional Maxwell equations. J. Comput. Phys. 109, 222–237 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Assous, F., Ciarlet, Jr, P., Labrunie, S.: Theoretical tools to solve the axisymmetric Maxwell equations. Math. Meth. Appl. Sci. 25, 49–78 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Assous, F., Ciarlet, Jr, P., Labrunie, S.: Solution of axisymmetric Maxwell equations. Math. Meth. Appl. Sci. 26, 861–896 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Assous, F., Ciarlet, Jr, P., Labrunie, S., Segré, J.: Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the Singular Complement Method. J. Comput. Phys. 191, 147–176 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Assous, F., Ciarlet, Jr, P., Sonnendrücker, E.: Resolution of the Maxwell equations in a domain with reentrant corners. Math. Model. Numer. Anal. 32, 359–389 (1998)

    MATH  MathSciNet  Google Scholar 

  7. Assous, F., Ciarlet, Jr., P., Segré, J.: Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: the Singular Complement Method. J. Comput. Phys. 161, 218–249 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Belhachmi, Z., Bernardi, C., Deparis, S.: Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem. Internal Report 03029, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie. (2003)

  9. Bernardi, C., Dauge, M., Maday, Y.: Polynomials in weighted Sobolev spaces: basics and trace liftings. Internal Report 92039, Laboratoire d'analyse numérique, Université Pierre et Marie Curie (1992)

  10. Bernardi, C., Dauge, M., Maday, Y.: Spectral methods for axisymmetric domains. Series in Applied Mathematics, Gauthier-Villars, Paris and North Holland, Amsterdam (1999)

  11. Ciarlet, Jr, P., He, J.: The singular complement method for 2D scalar problems. C. R. Acad. Sci. Paris, Ser. I 336, 353–358 (2003)

    MATH  MathSciNet  Google Scholar 

  12. Ciarlet, Jr, P., Jung, B., Kaddouri, S., Labrunie, S., Zou, J.: The Fourier Singular Complement Method for the Poisson problem. Part I: prismatic domains Numer. Math. 101, 423–450 (2005)

    MATH  MathSciNet  Google Scholar 

  13. Ciarlet, Jr, P., Jung, B., Kaddouri, S., Labrunie, S., Zou, J.: The Fourier Singular Complement Method for the Poisson problem. Part III: implementation issues. In preparation. (2004)

  14. Grisvard, P.: Singularities in boundary value problems. RMA 22, Masson, Paris (1992)

  15. Heinrich, B.: Singularity functions at axisymmetric edges and their representation by Fourier series. Math. Meth. Appl. Sci. 16, 837–854 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Heinrich, B.: The Fourier-finite element method for Poisson's equation in axisymmetric domains with edges. SIAM J. Numer. Anal. 33, 1885–1911 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Heinrich, B., Nicaise S., Weber, B.: Elliptic interface problems in axisymmetric domains II: Convergence analysis of the Fourier-finite element method. Adv. Math. Sci. Appl. 10, 571–600 (2000)

    MATH  MathSciNet  Google Scholar 

  18. Mercier, B., Raugel, G.: Résolution d'un problème aux limites dans un ouvert axisymétrique par éléments finis en r, z et séries de Fourier en θ. RAIRO Analyse numérique 4, 405–461 (1982)

    MathSciNet  Google Scholar 

  19. Nazarov, S.A., Plamenevsky, B.A.: Elliptic problems in domains with piecewise smooth boundaries. De Gruyter Exp. Math. 13 (1994)

  20. Raugel, G.: Résolution numérique de problèmes elliptiques dans des domaines avec coins. PhD Thesis, Rennes University, France (1978)

  21. Zienkiewicz, O.C.: The Finite Element Method. McGraw–Hill, London (1977)

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Authors and Affiliations

  1. ENSTA & CNRS UMR 2706, 32, boulevard Victor, 75739, Paris Cedex 15, France

    P. Ciarlet Jr. & S. Kaddouri

  2. Department of Mathematics, Chemnitz University of Technology, 09107, Chemnitz, Germany

    B. Jung

  3. IECN, Université Henri Poincaré Nancy I & INRIA (projet CALVI), 54506, Vandœuvre-lès- Nancy cedex, France

    S. Labrunie

  4. Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T, Hong Kong

    J. Zou

Authors
  1. P. Ciarlet Jr.
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  2. B. Jung
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  3. S. Kaddouri
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  4. S. Labrunie
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  5. J. Zou
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Ciarlet, P., Jung, B., Kaddouri, S. et al. The Fourier Singular Complement Method for the Poisson problem. Part II: axisymmetric domains. Numer. Math. 102, 583–610 (2006). https://doi.org/10.1007/s00211-005-0664-8

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  • DOI: https://doi.org/10.1007/s00211-005-0664-8

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