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Residual-based a posteriori error estimation for multipoint flux mixed finite element methods

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Abstract

A novel residual-type a posteriori error analysis technique is developed for multipoint flux mixed finite element methods for flow in porous media in two or three space dimensions. The derived a posteriori error estimator for the velocity and pressure error in \(L^{2}\)-norm consists of discretization and quadrature indicators, and is shown to be reliable and efficient. The main tools of analysis are a locally postprocessed approximation to the pressure solution of an auxiliary problem and a quadrature error estimate. Numerical experiments are presented to illustrate the competitive behavior of the estimator.

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References

  1. Aavatsmark, I., Barkve, T., Bøe, Ø., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. I. Derivation of the methods. SIAM J. Sci. Comput. 19, 1700–1716 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aavatsmark, I.: An introduction to multipoint flux approximations for quadrilateral grids. Comput. Geosci. 6, 405–432 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Achchab, B., Agouzal, A., Baranger, J., Maître, J.F.: Estimateur d’erreur a posteriori hiérarchique. Application aux éléments finis mixtes. Numer. Math. 80, 159–179 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ainsworth, M.: A synthesis of a posteriori error estimation techniques for conforming, nonconforming and discontinuous Galerkin finite element methods. In: Recent advances in adaptive computation, Contemp. Math. AMS, 383, pp 1–14. Providence, RI (2005)

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)

    Book  MATH  Google Scholar 

  7. Alonso, A.: Error estimators for a mixed method. Numer. Math. 74, 385–395 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica, pp 1–155 (2006)

  9. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. Babuška, I., Strouboulis, T.: The finite element method and its reliability. Oxford Science Publications (2001)

  11. 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 

  12. Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal. 33, 2431–2444 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brezzi, F., Douglas, J., Duran, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 794 54, 237–250 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite 797 elements for second order elliptic problems. Numer. Math. 47(798), 217–235 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Ser. Comput. Math. 15, Springer-Verlag, Berlin (1991)

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Carstensen, C., Bartels, S., Jansche, S.: A posteriori error estimates for nonconforming finite element methods. Numer. Math. 92, 233–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Carstensen, C., Hu, J., Orlando, A.: Framework for the a posteriori error analysis of nonconforming finite elements. SIAM J. Numer. Anal. 45, 68–82 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Nort-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  20. Du, S.H., Xie, X.P.: Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problem with Dirac delta source terms. Sci. Chin. Ser. A Math. 51, 1440–1460 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Du, S.H., Xie, X.P.: On residual-based posteriori error estimators for lowest-order Raviart–Thomas element approximation to convection–diffusion-reaction equations. J. Comput. Math. 32, 522–546 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Edwards, M.G.: Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput. Geosci. 6, 433–452 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  24. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, NJ (1985)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Kirby, R.: Residual a posteriori error estimates for the mixed finite element method. Comput. Geosci. 7, 197–214 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Klausen, R.A., Winther, R.: Robust convergence of multi point flux approximation on rough grids. Numer. Math. 104, 317–337 (2006). 842

    Article  MathSciNet  MATH  Google Scholar 

  28. Klausen, R.A., Winther, R.: Convergence of multipoint flux approximations on quadrilateral grids. Numer. Methods Partial Differ. Equ. 22, 1438–1454 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lovadina, C., Stenberg, R.: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comp. 75, 1659–1674 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. 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 

  32. Verfürth, R.: A review of posteriori error estimation and adaptive mesh-refinement techniques. Teubner Wiley, Stuttgart (1996)

    MATH  Google Scholar 

  33. Vohralík, M.: A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations. SIAM J. Numer. Anal. 45, 1570–1599 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Wheeler, M., Xue, G., Yotov, I.: A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra. Numer. Math. 121, 165–204 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wheeler, M., Xue, G., Yotov, I.: Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity. Comput. Geosci. Doi:10.1007/s10596-013-9382-y

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Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400047, China

    Shaohong Du

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

    Shaohong Du

  3. Computational Transport Phenomena Laboratory, The Physical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Kingdom of Saudi Arabia

    Shuyu Sun

  4. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Xiaoping Xie

Authors
  1. Shaohong Du
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  2. Shuyu Sun
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  3. Xiaoping Xie
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Correspondence to Xiaoping Xie.

Additional information

This work was supported by National Natural Science Foundation of China (11171239) and Major Research Plan of National Natural Science Foundation of China (91430105).

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Du, S., Sun, S. & Xie, X. Residual-based a posteriori error estimation for multipoint flux mixed finite element methods. Numer. Math. 134, 197–222 (2016). https://doi.org/10.1007/s00211-015-0770-1

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  • DOI: https://doi.org/10.1007/s00211-015-0770-1

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