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A posteriori error analysis of multipoint flux mixed finite element methods for interface problems

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Abstract

In this paper, the multipoint flux mixed finite element method is used to approximate the flux of two-dimensional elliptic interface problems. Within the class of modified quasi-monotonically distributed coefficients, we derive uniformly robust residual-type a posteriori error estimators for the flux error. Based on the residual-type estimator, we further develop robust implicit and explicit recovery-type estimators through gradient recovery in H(curl) conforming finite element spaces. Numerical experiments are presented to support the theoretical results.

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References

  1. Ainsworth, M.: Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42(6), 2320–2341 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ainsworth, M.: A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements. SIAM J. Sci. Comput. 30, 189–204 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85, 579–608 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fortin, F., Brezzi, M.: Mixed and hybrid finite element methods, vol. 15 of Springer Series in Computational Mathematics. Springer Verlag, Berlin (1991)

    MATH  Google Scholar 

  5. Cai, Z., Zhang, S.: Recovery-based error estimator for interface problems: Conforming linear elements. SIAM J. Numer. Anal. 47, 2132–2156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cai, Z., Zhang, S.: Recovery-based error estimators for interface problems: Mixed and nonconforming finite elements. SIAM J. Numer. Anal. 48, 30–52 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cai, Z., Zhang, S.: Robust equilibrated residual error estimator for diffusion problems: Conforming elements. SIAM J. Numer. Anal. 50(1), 151–170 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carstensen, C: A posteriori error estimate for the mixed finite method. Math. Comp. 66, 465–476 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ciarlet, P. G.: The finite element method for elliptic problems. Nort-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  10. Ciarlet, P. G.: Basic error estimates for elliptic problems. North-Holland, Amsterdam (1991)

    Book  MATH  Google Scholar 

  11. Demlow, A., Hirani, A.: A posteriori error estimates for finite elemnt exterior calculus: The deRham complex. Found. Comput. Math. 14(6), 1337–1371 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dryja, M., Sarkis, M. V., Widlund, O. B.: Multilevel Schwarz methods for elliptic probelms with discontinuous coefficients in three dimensions. Numer. Math. 72, 313–348 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Du, S.H.: A new residual posteriori error estimates of mixed finite element methods for convection-diffusion-reaction equations. Numer. meth. PDE 30, 593–624 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Du, S.H., Sun, S.Y., Xie, X.P.: Residual-based a posteriori error estimation for multipoint flux mixed finite element methods. Numer. Math. (2015). doi:10.1007/s00211-015-0770-1

    MathSciNet  Google Scholar 

  15. Girault, V., Raviart, P. A.: Finite element methods for Navier-Stokes equations. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  16. Luce, B.R., Wohlmuth, I.: A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42, 1394–1414 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nedelec, J.C.: Mixed finite eleemnts in \(\mathfrak {R}^{3}\). Numer. Math. 35, 315–341 (1980)

    Article  MathSciNet  Google Scholar 

  18. Nedelec, J.C.: A new family of mixed finite elements in \(\mathfrak {R}^{3}\). Numer. Math. 50, 57–81 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ovall, J.S.: Two Dangers to Avoid When Using Gradinet Recovery Methods for Finite Element Error Estimation and Adaptivity, Technical report, vol. 6. Max-Planck-Institute fur Mathematick in den Naturwissenschaften, Bonn, Germanny (2006)

    Google Scholar 

  20. OVALL, J. S.: Fixing a “Bug” in Recovery-type A Posteriori Error Estimators, Technical report, vol. 25. Max-Planck-Institute fur Mathematick in den Naturwissenschaften, Bonn, Germanny (2006)

    Google Scholar 

  21. Petzoldt, M: A posteriori error estimators for elliptic equations with discontinuous coeffcients. Adv. Comput. Math. 16, 47–75 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Raviart, R.A., Thomas, J.M.: A mixed finite element method for 2nd orderelliptic problems, in Mathematical aspects of the finite element method, lecture notes in mathematics, vol. 606, pp 292–315. Springer-Verlag, NewYork (1977)

    Google Scholar 

  23. Vohralík, M.: Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coeffients. J. Sci. Comput. 46(3), 397–438 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Verfürth, R: A posteriori error estimates and adaptive mesh-refinment techniques. J. Comput. Appl. Math. 50, 67–83 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Verfürth, R: A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comp. 62, 445–475 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Xu, J.: Counterexamples concerning a weighted L 2 projection. Math. Comp. 57, 563–568 (1991)

    MathSciNet  MATH  Google Scholar 

  27. Wheeler, M.F., Yotov, I.: A posteriori error estimates for the mortar mixed finite element method. SIAM J. Numer. Anal. 43, 1021–1042 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wheeler, M.F., Yotov, I.: A multipoint flux mixed finite element method. SIAM. J. Numer. Anal. 44, 2082–2106 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ingram, R., Wheeler, M.F., Yotov, I.: A multipoint flux mixed finite element method on hexahedra. SIAM J. Numer. Anal. 48, 1281–1312 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang, Z.: Recovery techniques in finite element methods. In: Tang, T., Xu, J. (eds.) Adaptive Computations: Theory and algorithms, Mathematics Monogr. Ser. 6, pp 333–412. Science Publisher, New York (2007)

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Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Shaohong Du

  2. Beijing Computational Science Research Center, Beijing, 100084, China

    Shaohong Du & Zhimin Zhang

  3. Department of Mathematics and Physics, Texas A&M International University, Laredo, TX, 78041, USA

    Runchang Lin

  4. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

Authors
  1. Shaohong Du
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  2. Runchang Lin
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  3. Zhimin Zhang
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Corresponding author

Correspondence to Shaohong Du.

Additional information

Communicated by: Martin Stynes

This research is supported in part by the National Natural Science Foundation of China under grants 91430216, 11471031, and 11428103, the US National Science Foundation under grants DMS-1419040 and DMS-1217268, and a University Research Grant of Texas A&M International University.

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Du, S., Lin, R. & Zhang, Z. A posteriori error analysis of multipoint flux mixed finite element methods for interface problems. Adv Comput Math 42, 921–945 (2016). https://doi.org/10.1007/s10444-015-9447-7

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  • DOI: https://doi.org/10.1007/s10444-015-9447-7

Keywords

Mathematics Subject Classfication (2010)