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Nitsche-mortaring for singularly perturbed convection–diffusion problems

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Abstract

In the present paper we analyse a finite element method for a singularly perturbed convection–diffusion problem with exponential boundary layers. Using a mortaring technique we combine an anisotropic triangulation of the layer region (into rectangles) with a shape regular one of the remainder of the domain. This results in a possibly non-matching (and hybrid), but layer adapted mesh of Shishkin type. We study the error of the method allowing different asymptotic behaviour of the triangulations and prove uniform convergence and a supercloseness property of the method. Numerical results supporting our analysis are presented.

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References

  1. Apel, T.: Anisotropic finite elements: Local estimates and applications. In: Advances in Numerical Mathematics. Teubner, Leipzig (1999)

    Google Scholar 

  2. Apel, T., Dobrowolski, M.: Anisotropic interpolation with applications to the finite element method. Computing 47(3–4), 277–293 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Becker, R., Hansbo, P., Stenberg, R.: A finite element method for domain decomposition with non-matching grids. Math. Model. Numer. Anal. 37(2), 209–225 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Braess, D., Dahmen, W., Wieners, C.: A multigrid algorithm for the mortar finite element method. SIAM J. Numer. Anal. 37(1), 48–69 (1999). doi:10.1137/S0036142998335431

    Article  MathSciNet  MATH  Google Scholar 

  6. Burman, E., Ern, A.: Continuous interior penalty hp-finite element methods for transport operators. In: de Castro, A.B., et al. (eds.) Numerical Mathematics and Advanced Applications. Proceedings of ENUMATH 2005, The 6th European Conference on Numerical Mathematics and Advanced Applications, Santiago de Compostela, Spain, 18–22 July 2005, pp. 504–511. Springer, Berlin (2006)

  7. Heinrich, B., Pönitz, K.: Nitsche type mortaring for singularly perturbed reaction–diffusion problems. Computing 75(4), 257–279 (2005). doi:10.1007/s00607-005-0123-5

    Article  MathSciNet  MATH  Google Scholar 

  8. Linß, T.: Uniform superconvergence of a Galerkin finite element method on Shishkin-type meshes. Numer. Methods Partial Differ. Equ. 16(5), 426–440 (2000)

    Article  MATH  Google Scholar 

  9. Linß, T.: Layer-adapted meshes for reaction-convection–diffusion problems. In: Lecture Notes in Mathematics, vol. 1985. Springer, Berlin (2010)

    Google Scholar 

  10. Linß, T., Stynes, M.: Asymptotic analysis and Shishkin-type decomposition for an elliptic convection–diffusion problem. J. Math. Anal. Appl. 261, 604–632 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer, Berlin (1972)

    MATH  Google Scholar 

  12. Melenk, J.M.: hp-finite element methods for singular perturbations. In: Lecture Notes in Mathematics, vol. 1796. Springer, Berlin (2002)

    Google Scholar 

  13. Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Fitted numerical methods for singular perturbation problems. In: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions. World Scientific, Singapore (1996)

    MATH  Google Scholar 

  14. Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hamb. 36, 9–15 (1971). In German

    Article  MathSciNet  MATH  Google Scholar 

  15. Pönitz, K.: Finite-Elemente-Mortaring nach einer Methode von J. A. Nitsche für elliptische Randwertaufgaben. Ph.D. thesis, Technische Universität Chemnitz (2006). In German

  16. Roos, H.G.: Superconvergence on a hybrid mesh for singularly perturbed problems with exponential layers. ZAMM, Z. Angew. Math. Mech. 86(8), 649–655 (2006). doi:10.1002/zamm.200510264

    Article  MathSciNet  MATH  Google Scholar 

  17. Roos, H.G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations. In: Convection–Diffusion–Reaction and Flow Problems, 2nd ed., vol. 24. Springer, Berlin (2008)

    Google Scholar 

  18. Schieweck, F.: On the role of boundary conditions for CIP stabilization of higher order finite elements. ETNA, Electron. Trans. Numer. Anal. 32, 1–16 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Stynes, M., O’Riordan, E.: A uniformly convergent Galerkin method on a Shishkin mesh for a convection–diffusion problem. J. Math. Anal. Appl. 214(1), 36–54 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Stynes, M., Tobiska, L.: The SDFEM for a convection–diffusion problem with a boundary layer: optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal. 41(5), 1620–1642 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Warburton, T., Hesthaven, J.S.: On the constants in hp-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Eng. 192(25), 2765–2773 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, Z.: Finite element superconvergence on Shishkin mesh for 2-d convection–diffusion problems. Math. Comput. 72(243), 1147–1177 (2003)

    Article  MATH  Google Scholar 

  23. Zienkiewicz, O., Zhu, J.: Superconvergence and the superconvergent patch recovery. Finite Elem. Anal. Des. 19(1–2), 11–23 (1995)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Numerische Mathematik, TU Dresden, 01062, Dresden, Germany

    Torsten Linß, Hans-Görg Roos & Martin Schopf

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  1. Torsten Linß
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  2. Hans-Görg Roos
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  3. Martin Schopf
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Correspondence to Torsten Linß.

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Communicated by Martin Stynes.

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Linß, T., Roos, HG. & Schopf, M. Nitsche-mortaring for singularly perturbed convection–diffusion problems. Adv Comput Math 36, 581–603 (2012). https://doi.org/10.1007/s10444-011-9195-2

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  • DOI: https://doi.org/10.1007/s10444-011-9195-2

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