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BIT Numerical Mathematics

, Volume 53, Issue 3, pp 791–820 | Cite as

A uniformly well-conditioned, unfitted Nitsche method for interface problems

  • Eddie WadbroEmail author
  • Sara Zahedi
  • Gunilla Kreiss
  • Martin Berggren
Article

Abstract

A finite element method for elliptic partial differential equations that allows for discontinuities along an interface not aligned with the mesh is presented. The solution on each side of the interface is separately expanded in standard continuous, piecewise-linear functions, and jump conditions at the interface are weakly enforced using a variant of Nitsche’s method. In our method, the solutions on each side of the interface are extended to the entire domain which results in a fixed number of unknowns independent of the location of the interface. A stabilization procedure is included to ensure well-defined extensions. We prove that the method provides optimal convergence order in the energy and the L 2 norms and a condition number of the system matrix that is independent of the position of the interface relative to the mesh. Numerical experiments confirm the theoretical results and demonstrate optimal convergence order also for the pointwise errors.

Keywords

Interface problem Nitsche’s method Interior penalties Finite element methods 

Mathematics Subject Classification (2000)

35J05 35J20 65N15 65N30 

References

  1. 1.
    Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Babuska, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207–213 (1970) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Belytschko, T., Moes, N., Usui, S., Parimi, C.: Arbitrary discontinuitites in finite elements. Int. J. Numer. Methods Eng. 50, 993–1013 (2001) zbMATHCrossRefGoogle Scholar
  4. 4.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (2008) zbMATHCrossRefGoogle Scholar
  5. 5.
    Burman, E.: Ghost penalty. C. R. Acad. Sci. Paris Sér. I Math. 348, 1217–1220 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 64(4), 328–341 (2012) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79, 175–202 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chessa, J., Belytschko, T.: An extended finite element method for two-phase fluids. J. Appl. Mech. 70, 10–17 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dautray, R., Lions, J.L.: Functional and Variational Methods. Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2. Springer, Berlin (1988) zbMATHGoogle Scholar
  10. 10.
    Fries, T.P., Belytschko: The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int. J. Numer. Methods Eng. 68, 1358–1385 (2006) zbMATHCrossRefGoogle Scholar
  11. 11.
    Grisvard, P.: Elliptic Problems in Non-smooth Domains. Pitman, London (1985) Google Scholar
  12. 12.
    Gross, S., Reusken, A.: An extended pressure finite element space for two-phase incompressible flows with surface tension. J. Comput. Phys. 45, 40–58 (2007) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191, 5537–5552 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Huang, J., Zou, J.: A mortar element method for elliptic problems with discontinuous coefficients. IMA J. Numer. Anal. 22, 549–576 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Huh, J.S., Sethian, J.A.: Exact subgrid interface correction schemes for elliptic interface problems. Proc. Natl. Acad. Sci. USA 105(29), 9874–9879 (2008) zbMATHCrossRefGoogle Scholar
  16. 16.
    Joseph, D.D., Renardy, Y.Y.: Fundamentals of Two-Fluid Dynamics. Springer, New York (1993) Google Scholar
  17. 17.
    Lamichhane, B.P., Wohlmuth, B.I.: Mortar finite elements for interface problems. Computing 72, 333–348 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Li, Z., Lin, T., Lin, Y., Rogers, R.C.: An immersed finite element space and its approximation capability. Numer. Methods Partial Differ. Equ. 20(3), 338–367 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Li, Z., Lin, T., Wu, X.: New Cartesian grid methods for interface problems using the finite element formulation. Numer. Math. 96(1), 61–98 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (2008) zbMATHGoogle Scholar
  22. 22.
    Reusken, A.: Analysis of extended pressure finite element space for two-phase incompressible flows. Comput. Vis. Sci. 11, 293–305 (2008) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Tornberg, A.K., Engquist, B.: Regularization techniques for numerical approximation of PDEs with singularities. J. Sci. Comput. 19, 527–552 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Wadbro, E., Zahedi, S., Kreiss, G., Berggren, M.: A uniformly well-conditioned, unfitted Nitsche method for interface problems: Part II. TRITA-NA 2011:2, KTH/NA-11/02-SE. ISSN 0348-2952 (2011) Google Scholar
  25. 25.
    Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987) zbMATHCrossRefGoogle Scholar
  26. 26.
    Zahedi, S.: Numerical methods for fluid interface problems. Ph.D. thesis, KTH, School of Computer Science and Communication, KTH Royal Institute of Technology (2011) Google Scholar
  27. 27.
    Zahedi, S., Tornberg, A.K.: Delta function approximations in level set methods by distance function extension. J. Comput. Phys. 229, 2199–2219 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Zahedi, S., Wadbro, E., Kreiss, G., Berggren, M.: A uniformly well-conditioned, unfitted Nitsche method for interface problems: Part I. TRITA-NA 2011:1, KTH/NA-11/01-SE. ISSN 0348-2952 (2011) Google Scholar
  29. 29.
    Zunino, P., Cattaneo, L., Colciago, C.M.: An unfitted interface penalty method for the numerical approximation of contrast problems. Appl. Numer. Math. 61(10), 1059–1076 (2011) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Eddie Wadbro
    • 1
    Email author
  • Sara Zahedi
    • 2
  • Gunilla Kreiss
    • 2
  • Martin Berggren
    • 1
  1. 1.Department of Computing ScienceUmeå UniversityUmeåSweden
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden

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