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Optimal Finite Element Methods for Interface Problems

  • Jinchao XuEmail author
  • Shuo Zhang
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)

Abstract

There are many physical problems such as multiphase flows and fluid-structure interactions whose solutions are piecewise smooth but may have discontinuity across some curved interfaces. The direct application of standard finite element method may not perform well. In this paper, we study some special finite element methods for this type of problems. For simplicity of exposition, we consider the case that there is only one interface which is smooth. Let \(\varOmega,\varOmega _{1} \subset \mathbb{R}^{2}\) be two bounded domains with \(\varOmega _{1} \subset \varOmega\). We assume that Γ = ∂ Ω1 is sufficiently smooth, and \(\varGamma \cap \partial \varOmega =\emptyset\). To be focused on the influence of Γ, we assume \(\varOmega = (-1,1)^{2}\).

Keywords

Element Space Multigrid Method Linear Element Interpolation Operator Piecewise Smooth Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgement

The authors would like to thank Dr. Xiaozhe Hu for his help on the numerical examples.

References

  1. 1.
    G. Acosta, R.G. Durán, The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal. 37(1), 18–36 (1999) (electronic). ISSN 0036-1429. doi:10.1137/S0036142997331293. http://dx.doi.org/10.1137/S0036142997331293
  2. 2.
    J.M. Arrieta, A. Rodríguez-Bernal, J.D. Rossi, The best Sobolev trace constant as limit of the usual Sobolev constant for small strips near the boundary. Proc. R. Soc. Edinb. Sect. A 138(2), 223–237 (2008). ISSN 0308-2105. doi:10.1017/S0308210506000813. http://dx.doi.org/10.1017/S0308210506000813
  3. 3.
    J.H. Bramble, J.T. King, A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6(2), 109–138 (1996). ISSN 1019-7168. doi:10.1007/BF02127700. http://dx.doi.org/10.1007/BF02127700
  4. 4.
    F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15 (Springer, New York, 1991). ISBN 0-387-97582-9. doi:10.1007/978-1-4612-3172-1. http://dx.doi.org/10.1007/978-1-4612-3172-1
  5. 5.
    L. Chen, R.H. Nochetto, J. Xu, Optimal multilevel methods for graded bisection grids. Numer. Math. 120(1), 1–34 (2012). ISSN 0029-599X. doi:10.1007/s00211-011-0401-4. http://dx.doi.org/10.1007/s00211-011-0401-4
  6. 6.
    Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79(2), 175–202 (1998). ISSN 0029-599X. doi:10.1007/s002110050336. http://dx.doi.org/10.1007/s002110050336.
  7. 7.
    Z. Chen, Z. Wu, Y. Xiao, An adaptive immersed finite element method with arbitrary lagrangian-eulerian scheme for parabolic equations in variable domains. Int J. Numer. Anal. Model. 12(3), 567–591 (2015)MathSciNetGoogle Scholar
  8. 8.
    J. Li, J.M. Melenk, B. Wohlmuth, J. Zou, Optimal a priori estimates for higher order finite elements for elliptic interface problems. Appl. Numer. Math. 60(1–2), 19–37 (2010). ISSN 0168-9274. doi:10.1016/j.apnum.2009.08.005. http://dx.doi.org/10.1016/j.apnum.2009.08.005
  9. 9.
    L.R. Scott, S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990). ISSN 0025-5718. doi:10.2307/2008497. http://dx.doi.org/10.2307/2008497
  10. 10.
    J. Xu, Estimate of the convergence rate of finite element solutions to elliptic equations of second order with discontinuous coefficients (in chinese). Natural Science Journal of Xiangtan University (1982), pp. 1–5Google Scholar
  11. 11.
    J. Xu, Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992) ISSN 0036-1445. doi:10.1137/1034116. http://dx.doi.org/10.1137/1034116
  12. 12.
    J. Xu, Estimate of the convergence rate of finite element solutions to elliptic equations of second order with discontinuous coefficients. arXiv preprint arXiv:1311.4178 (2013)Google Scholar
  13. 13.
    J. Xu, Y. Zhu, Uniform convergent multigrid methods for elliptic problems with strongly discontinuous coefficients. Math. Models Methods Appl. Sci. 18(1), 77–105 (2008) ISSN 0218-2025. doi:10.1142/S0218202508002619. http://dx.doi.org/10.1142/S0218202508002619
  14. 14.
    J. Xu, L. Zikatanov, The method of alternating projections and the method of subspace corrections in Hilbert space. J. Am. Math. Soc. 15(3), 573–597 (2002). ISSN 0894-0347. doi:10.1090/S0894-0347-02-00398-3. http://dx.doi.org/10.1090/S0894-0347-02-00398-3
  15. 15.
    J. Xu, L. Chen, R.H. Nochetto, Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids, in Multiscale, Nonlinear and Adaptive Approximation (Springer, Berlin, 2009), pp. 599–659. doi:10.1007/978-3-642-03413-8_14. http://dx.doi.org/10.1007/978-3-642-03413-8_14 zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Center for Computational Mathematics and ApplicationsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.LSEC, ICMSEC, NCMIS, Academy of Mathematics and System SciencesChinese Academy of SciencesBeijingChina

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