Recovery of the Interface Velocity for the Incompressible Flow in Enhanced Velocity Mixed Finite Element Method

  • Yerlan AmanbekEmail author
  • Gurpreet Singh
  • Mary F. Wheeler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11539)


The velocity, coupling term in the flow and transport problems, is important in the accurate numerical simulation or in the posteriori error analysis for adaptive mesh refinement. We consider Enhanced Velocity Mixed Finite Element Method (EVMFEM) for the incompressible Darcy flow. In this paper, our aim is to study the improvement of velocity at interface to achieve the better approximation of velocity between subdomains. We propose the reconstruction of velocity at interface by using the post-processed pressure. Numerical results at the interface show improvement on convergence rate.


Domain decomposition Enhanced Velocity Velocity improvement 



First author would like to thank Drs. I. Yotov and T. Arbogast for discussions on formulation of the different view of EVMFEM. This research is supported by Faculty Development Competitive Research Grant (Grant No. 110119FD4502), Nazarbayev University.


  1. 1.
    Ainsworth, M.: Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42(6), 2320–2341 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amanbek, Y.: A new adaptive modeling of flow and transport in porous media using an enhanced velocity scheme. Ph.D. thesis (2018)Google Scholar
  3. 3.
    Amanbek, Y., Singh, G., Wheeler, M.F., van Duijn, H.: Adaptive numerical homogenization for upscaling single phase flow and transport. J. Comput. Phys. 387, 117–133 (2019)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Amanbek, Y., Wheeler, M.: A priori error analysis for transient problems using Enhanced Velocity approach in the discrete-time setting. arXiv preprint arXiv:1812.04809 (2018)
  5. 5.
    Arbogast, T., Chen, Z.: On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comput. 64(211), 943–972 (1995)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Arbogast, T., Estep, D., Sheehan, B., Tavener, S.: A posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with nonmatching grids. IMA J. Numer. Anal. 34(4), 1625–1653 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Burman, E., Ern, A.: Continuous interior penalty \(hp\)-finite element methods for advection and advection-diffusion equations. Math. Comput. 76(259), 1119–1140 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media, vol. 2. SIAM, New Delhi (2006)CrossRefGoogle Scholar
  9. 9.
    Gerritsen, M., Lambers, J.: Integration of local-global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations. Comput. Geosci. 12(2), 193–208 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Glowinski, R., Wheeler, M.F.: Domain decomposition and mixed finite element methods for elliptic problems. In: First International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp. 144–172 (1988)Google Scholar
  11. 11.
    Karakashian, O.A., Pascal, F.: A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41(6), 2374–2399 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pencheva, G.V., Vohralík, M., Wheeler, M.F., Wildey, T.: Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling. SIAM J. Numer. Anal. 51(1), 526–554 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media, pp. 35–106. SIAM (1983)Google Scholar
  14. 14.
    Singh, G., Amanbek, Y., Wheeler, M.F.: Adaptive homogenization for upscaling heterogeneous porous medium. In: SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2017)Google Scholar
  15. 15.
    Thomas, S.G., Wheeler, M.F.: Enhanced velocity mixed finite element methods for modeling coupled flow and transport on non-matching multiblock grids. Comput. Geosci. 15(4), 605–625 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Vohralík, M.: Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods. Math. Comput. 79(272), 2001–2032 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wheeler, J.A., Wheeler, M.F., Yotov, I.: Enhanced velocity mixed finite element methods for flow in multiblock domains. Comput. Geosci. 6(3–4)Google Scholar
  18. 18.
    Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineerng analysis. Int. J. Numer. Meth. Eng. 24(2), 337–357 (1987)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Yerlan Amanbek
    • 1
    • 2
    Email author
  • Gurpreet Singh
    • 1
  • Mary F. Wheeler
    • 1
  1. 1.Center for Subsurface Modeling, Oden Institute for Computational Engineering and SciencesUniversity of Texas at AustinAustinUSA
  2. 2.Nazarbayev UniversityAstanaKazakhstan

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