SpringerLink
Log in

Reduced-order multiscale modeling of nonlinear p-Laplacian flows in high-contrast media

  • ORIGINAL PAPER
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

In this paper, we consider a class of nonlinear flow problems that are modeled through a time-dependent p-Laplacian formulation. Using an approach that combines the Generalized Multiscale Finite Element Method (GMsFEM) and Discrete Empirical Interpolation Method (DEIM), we are able to accurately approximate the nonlinear p-Laplacian solutions at a significantly reduced cost. In particular, GMsFEM allows us to iteratively solve the global problem using a reduced-order model in which a flexible number of multiscale basis functions are used to construct a spectrally enriched coarse-grid solution space. The iterative procedure requires a number of nonlinear functional updates that are simultaneously made more cost efficient through implementation of DEIM. The combined GMsFEM-DEIM approach is shown to be a flexible framework for producing accurate reduced-order descriptions of the nonlinear model for a wide range of parameters and varying levels nonlinearity. A number of numerical examples are presented to illustrate the effectiveness of the proposed methodology.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aarnes, J., Krogstad, S., Lie, K.: A hierarchical multiscale method for two-phase flow based on upon mixed finite elements and nonuniform coarse grids. SIAM Multiscale Model. Simul. 5, 337–363 (2006)

    Article  Google Scholar 

  2. Ahmed, N., Sunada, D.: Nonlinear flow in porous media, vol. 95, pp 1847–1857 (1969)

  3. Alotaibi, M., Calo, V., Efendiev, Y., Galvis, J., Ghommem, M.: Global-local nonlinear model reduction for flows in heterogeneous porous media. ar**v:1407.0782 (2014)

  4. Arbogast, T., Pencheva, G., Wheeler, M., Yotov, I.: A multiscale mortar mixed finite element method. SIAM Multiscale Model. Simul. 6, 319–346 (2007)

    Article  Google Scholar 

  5. Astrita, G., Marrucci, G.: Principles of non-newtonian fluid mechanics. McGraw-Hill, New York (1974)

    Google Scholar 

  6. Barrault, M., Maday, Y., Nguyen, N., Patera, A.: An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339 (2004)

  7. Barrett, J., Liu, W.: Finite element approximation of the p-Laplacian. Math. Comput. 61(204), 523–537 (1993)

    Google Scholar 

  8. Bazilevs, Y., Calo, V.M., Cottrell, J.A., Hughes, T.J.R., Reali, A., Scovazzi, G.: Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput. Methods Appl. Mech. Eng. 197, 173–201 (2007)

    Article  Google Scholar 

  9. Berkooz, G., Holmes, P., Lumley, J.L.: The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech. 53, 321–575 (1993)

    Google Scholar 

  10. Bognar, G.: Numerical and analytic investigation of some nonlinear problems in fluid mechanics. Computer and Simulation in Modern Science 2, 172–179 (2008)

    Google Scholar 

  11. Bonforte, M., Grillo, G.: Ultracontractive bounds for nonlinear evolution equations governed by the subcritical p-Laplacian. Progress in Nonlinear Differential Equations and Their Applications 61, 15–28 (2005)

    Google Scholar 

  12. Bush, L., Ginting, V., Presho, M.: Application of a conservative, generalized multiscale finite element method to flow models. J. Comput. Appl. Math. 260, 395–409 (2014)

    Article  Google Scholar 

  13. Butcher, J.: Numerical methods for ordinary differential equations. John Wiley and Sons, New York (2003)

    Book  Google Scholar 

  14. Calo, V., Efendiev, Y., Galvis, J., Ghommem, M.: Multiscale empirical interpolation for solving nonlinear pdes using generalized multiscale finite element methods, submitted to Journal of Computational Physics (2014). ar**v:1407.0103

  15. Calo, V.M., Efendiev, Y., Galvis, J.: A note on variational multiscale methods for high-contrast heterogeneous porous media flows with rough source terms. Adv. Water Resour. 9, 1177–1185 (2011)

    Article  Google Scholar 

  16. Chaturantabut, S., Sorensen, D.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764 (2010)

    Article  Google Scholar 

  17. Chu, J., Efendiev, Y., Ginting, V., Hou, T.: Flow based oversampling technique for multiscale finite element methods. Adv. Water Resour. 31, 599–608 (2008)

    Article  Google Scholar 

  18. Chung, E., Efendiev, Y., Li, G.: An adaptive gmsfem for high-contrast flow problems. J. Computat. Phys. 273, 54–76 (2014)

    Article  Google Scholar 

  19. Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34, A937–A969 (2012)

    Article  Google Scholar 

  20. Efendiev, Y., Galvis, J., Gildin, E.: Local-global multiscale model reduction for flows in highly heterogeneous media. J. Comput. Phys. 231, 8100–8113 (2012)

    Article  Google Scholar 

  21. Efendiev, Y., Galvis, J., Li, G., Presho, M.: Generalized multiscale finite element methods, nonlinear elliptic equations. Commun. Comput. Phys. 15 (2014)

  22. Efendiev, Y., Galvis, J., Presho, M., Zhou, J.: A multiscale enrichment procedure for nonlinear monotone operators. ESAIM: Math. Model. Numer. Anal. 48, 475–491 (2014)

    Article  Google Scholar 

  23. Efendiev, Y., Galvis, J., Wu, X.: Multiscale finite element methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230, 937–955 (2011)

    Article  Google Scholar 

  24. Efendiev, Y., Ginting, V., Hou, T., Ewing, R.: Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220, 155–174 (2006)

    Article  Google Scholar 

  25. Efendiev, Y., Hou, T.: Multiscale finite element methods: Theory and applications Surveys and tutorials in the applied mathematical sciences, vol. 4. Springer, New York (2009)

    Google Scholar 

  26. Elfverson, D., Ginting, V., Henning, P.: On multiscale methods in petrov-galerkin formulation. Numer. Math. (2015). doi:10.1007/s00211-015-0703-z

  27. Galvis, J., Efendiev, Y.: Domain decomposition preconditioners for multiscale flows in high contrast media. SIAM Multiscale Model. Simul. 8, 1461–1483 (2010)

    Article  Google Scholar 

  28. Galvis, J., Efendiev, Y.: Domain decomposition preconditioners for multiscale flows in high contrast media. Reduced dimension coarse spaces. SIAM Multiscale Model. Simul. 8, 1621–1644 (2010)

    Article  Google Scholar 

  29. Ghommem, M., Presho, M., Calo, V., Efendiev, Y.: Mode decomposition methods for flows in high-contrast porous media. Global-local approach. J. Comput. Phys. 253, 226–238 (2013)

    Article  Google Scholar 

  30. Hou, T., Wu, X.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134, 169–189 (1997)

    Article  Google Scholar 

  31. Ju, N.: Numerical analysis of parabolic p-laplacian: Approximation of trajectories. SIAM J. Numer. Anal. 37(6), 1861–1884 (2000)

    Article  Google Scholar 

  32. Kaulmann, S., Flemisch, B., Haasdonk, B., Lie, K., Ohlberger, M.: The localized reduced basis multiscale method for two-phase flows in porous media. ar**v:1405.2810 (2014)

  33. Kreuzer, C.: Reliable and efficient a posteriori error estimates for finite element approximations of the parabolic p-Laplacian. Calcolo 50, 79–110 (2013)

    Article  Google Scholar 

  34. Mastorakis, N., Fathabadi, H.: On the solution of p-Laplacian for non-Newtonian fluid flow, vol. 8, pp 238–245 (2009)

  35. Schmid, P.J., Meyer, K.E., Pust, O.: Dynamic mode decomposition and proper orthogonal decomposition of flow in a lid-driven cylindrical cavity. In: 8th international symposium on particle image velocimetry - PIV09 (2009)

  36. Wang, Z., Akhtar, I., Borggaard, J., Iliescu, T.: Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison. Comput. Methods Appl. Mech. Eng. 237–240, 10–26 (2012)

    Article  Google Scholar 

  37. Zeidler, E.: Nonlinear functional analysis and its applications III. Springer, New York (1985)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Computational and Engineering Sciences (ICES), University of Texas at Austin, Austin, Texas, USA

    M. Presho

  2. Department of Mathematics, Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas, USA

    S. Ye

Authors
  1. M. Presho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Presho.

Additional information

This material is based upon work supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-SC0009286 as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Presho, M., Ye, S. Reduced-order multiscale modeling of nonlinear p-Laplacian flows in high-contrast media. Comput Geosci 19, 921–932 (2015). https://doi.org/10.1007/s10596-015-9504-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-015-9504-9

Keywords

Search

Navigation