Skip to main content
SpringerLink
Log in

Optimal convergence analysis of an immersed interface finite element method

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We analyze an immersed interface finite element method based on linear polynomials on noninterface triangular elements and piecewise linear polynomials on interface triangular elements. The flux jump condition is weakly enforced on the smooth interface. Optimal error estimates are derived in the broken H 1-norm and L 2-norm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Babuska, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207–213 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bramble, J.H., King, J.T.: A finite element method for interface problems in domains with smooth boundary and interfaces. Adv. Comput. Math. 6, 109–138 (1996)

    Article  MathSciNet  Google Scholar 

  3. Bramble, J.H., Pasciak, J.E., Schatz, A.H.: The construction of preconditioners for elliptic problems by substructuring. III. Math. Comp. 51, 415–430 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)

    MATH  Google Scholar 

  5. Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79, 175–202 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chou, S.H., Kwak, D.Y.: A covolume method based on rotated bilinears for the generalized Stokes problem. SIAM J. Numer. Anal. 35, 494–507 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)

    MATH  Google Scholar 

  8. Ciarlet, P.G., Raviart, P.A.: General Lagrange and Hermite interpolation in ℝn with applications to finite element methods. Arch. Rational Mech. Anal. 46, 177–199 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  9. Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO. Anal. Numer. 7, 33–75 (1973)

    MathSciNet  Google Scholar 

  10. Douglas Jr., J., Santos, J.E., Sheen, D., Ye, X.: Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. M2AN 33, 747–770 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Efendiev, Y.R., Wu, X.H.: Multiscale finite element for problems with highly oscillatory coefficients. Numer. Math. 90, 459–486 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations. In: Theory and Algorithms. Springer, Berlin (1986)

    Google Scholar 

  13. Gong, Y., Li, B., Li, Z.: Immersed-interface finite-element methods for elliptic interface problems with non-homogeneous jump conditions. SIAM J. Numer. Anal. 46, 472–495 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Grisvard, P.: Elliptic problems in nonsmooth domains. In: Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985)

    Google Scholar 

  15. Hou, T., Li, Z., Osher, S., Zhao, H.: A hybrid method for moving interface problems with application to the Hele-Shaw flow. J. Comput. Phys. 134, 236–252 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Khoromskij, B.N., Wittum, G.: Numerical Solution of Elliptic Differential Equations by Reduction to the Interface. Springer, New York (2004)

    MATH  Google Scholar 

  17. Ladyzhenskaya, O.A., Rivkind, V. Ja., Ural’ceva, N.N.: The classical solvability of diffraction problems. In: Trudy Mat. Inst. Steklov, vol. 92, pp. 116–146. Translated in Proceedings of the Steklov Institute of Math., no. 92, Boundary Value Problems of Mathematical Physics IV. American Mathematical Society, Providence (1966)

  18. LeVeque, R.J., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. LeVeque, R.J., Li, Z.: Immersed interface method for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. 18, 709–735 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. LeVeque, R.J., Zhang, C.: Immersed interface methods for wave equations with discontinuous coefficients. Wave Motion 25, 237–263 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Li, Z.: Immersed interface method for moving interface problems. Numer. Algorithms 14, 269–293 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. Li, Z.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35, 230–254 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  23. Li, Z., Ito, K.: The Immnersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, p. 176. SIAM, Philadelphia (2006)

    Google Scholar 

  24. Li, Z., Lin, T., Lin, Y., Rogers, R.C.: An immersed finite element space and its approximation capability. Numer. Methods Partial Differential Equations 20, 338–367 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Li, Z., Lin, T., Wu, X.: New Cartesian grid methods for interface problems using the finite element formulation. Numer. Math. 96, 61–98 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lin, T., Lin, Y., Rogers, R., Ryan, M.L.: A rectangular immersed finite element space for interface problems. Adv. Comput. Theory Pract. 7, 107–114 (2001)

    MathSciNet  Google Scholar 

  27. Ren, X., Wei, J.: On a two-dimensional elliptic problem with large exponent in nonlinearity. Trans. Amer. Math. Soc. 343(2), 749–763 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Roitberg, J.A., Seftel, Z.G.: A theorem on homeomorphisms for elliptic systems and its applications. Math. USSR-Sb. 7, 439–465 (1969)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH, 43403-0221, USA

    So-Hsiang Chou

  2. Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea

    Do Y. Kwak & K. T. Wee

Authors
  1. So-Hsiang Chou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Do Y. Kwak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. T. Wee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to So-Hsiang Chou.

Additional information

Communicated by Martin Stynes.

The work of Do Y. Kwak was supported by KOSEF R01-2007-000-10062-0.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chou, SH., Kwak, D.Y. & Wee, K.T. Optimal convergence analysis of an immersed interface finite element method. Adv Comput Math 33, 149–168 (2010). https://doi.org/10.1007/s10444-009-9122-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-009-9122-y

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation