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On the connection between the stabilized Lagrange multiplier and Nitsche’s methods

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Abstract

We derive Nitsche’s method for the domain decomposition through the stabilized Lagrange multiplier method. Taking material parameters carefully into account this derivation naturally introduces parameter weighted average flux and stabilizing terms to Nitsche’s method. We show stability and a priori analyses in the mesh dependent norms for both the stabilized method and Nitsche’s method, and discuss connections between the proposed methods.

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References

  1. Annavarapu, C., Hautefeuille, M., Dolbow, J.E.: A robust Nitsches formulation for interface problems. Comput. Methods Appl. Mech. Eng. 225–228, 44–54 (2012)

    Article  MathSciNet  Google Scholar 

  2. Babuška, I.: The finite element method with Lagrangian multipliers. Numerische Mathematik 20, 179–192 (1973)

    Article  MATH  Google Scholar 

  3. Barbosa, H.J., Hughes, T.J.: The finite element method with Lagrange multipliers on the boundary: circumventing the Babuška–Brezzi condition. Comput. Methods Appl. Mech. Eng. 85(1), 109–128 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barbosa, H.J., Hughes, T.J.: Boundary Lagrange multipliers in finite element methods: error analysis in natural norms. Numerische Mathematik 62(1), 1–15 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barrau, N., Becker, R., Dubach, E., Luce, R.: A robust variant of NXFEM for the interface problem. Comptes Rendus Mathematique 350(15–16), 789–792 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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  MATH  MathSciNet  Google Scholar 

  7. Brenner, S.C., Scott, R.L.: The mathematical theory of finite element methods. In: Applied Mathematics, 2nd edn, vol. 15. Springer (2002)

  8. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R-2), pp. 129–151 (1974)

  9. Burman, E.: A penalty-free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions. SIAM J. Numer. Anal. 50(4), 1959–1981 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Burman, E.: Projection stabilization of Lagrange multipliers for the imposition of constraints on interfaces. Numer. Methods Partial Differ. Equ. 30(2), 567–592 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Burman, E., Zunino, P.: A domain decomposition method based on weighted interior penalties for advection–diffusion–reaction problems. SIAM J. Numer. Anal. 44(4), 1612–1638 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chu, C.C., Graham, I.G., Hou, T.Y.: A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79(272), 1915–1915 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications, vol. 69. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  14. Dryja, M.: On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients. Comput. Methods Appl. Math. 3(1), 76–85 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dupire, G., Boufflet, J., Dambrine, M., Villon, P.: On the necessity of Nitsche term. Appl. Numer. Math. 60(9), 888–902 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ern, A., Stephansen, A.F., Zunino, P.: A discontinuous Galerkin method with weighted averages for advection–diffusion equations with locally small and anisotropic diffusivity. IMA J. Numer. Anal. 29(2), 235–256 (2008)

    Article  MathSciNet  Google Scholar 

  17. Hansbo, P., Lovadina, C., Perugia, I., Sangalli, G.: A Lagrange multiplier method for the finite non-matching meshes. Numerische Mathematik 100, 91–115 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Huang, J., Zou, J.: Some new a priori estimates for second-order elliptic and parabolic interface problems. J. Differ. Equ. 184(2), 570–586 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Juntunen, M., Stenberg, R.: Nitsche’s method for general boundary conditions. Math. Comput. 78(267), 1353–1374 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  20. Plum, M., Wieners, C.: Optimal a priori estimates for interface problems. Numerische Mathematik 95, 735–759 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Stenberg, R.: On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63(1–3), 139–148 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Stenberg, R.: Mortaring by a method of J.A. Nitsche. Computational mechanics (Buenos Aires, 1998), Centro Internac. Métodos Numér. Ing., Barcelona (1998)

  23. Zunino, P., Cattaneo, L., Colciago, C.M.: An unfitted interface penalty method for the numerical approximation of contrast problems. Applied Numerical Mathematics 61(10), 1059–1076 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076, Aalto, Finland

    Mika Juntunen

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  1. Mika Juntunen
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Correspondence to Mika Juntunen.

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Juntunen, M. On the connection between the stabilized Lagrange multiplier and Nitsche’s methods. Numer. Math. 131, 453–471 (2015). https://doi.org/10.1007/s00211-015-0701-1

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  • DOI: https://doi.org/10.1007/s00211-015-0701-1

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