Skip to main content
SpringerLink
Log in

Generalized Multiscale Discontinuous Galerkin Method for Helmholtz Problem in Fractured Media

  • Conference paper
  • First Online:
Finite Difference Methods. Theory and Applications (FDM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11386))

Included in the following conference series:

  • 1358 Accesses

Abstract

In this work, we consider wave propagation in fractured media. The mathematical model is described by Helmholtz problem related to wave propagation with specific interface conditions on the fracture in the frequency domain. We use a discontinuous Galerkin method for the approximation by space that help to weakly impose interface conditions on fractures. Such approximations lead to the large system of equations and computationally expensive. In this work, we construct a coarse grid approximation for effective solution using Generalized Multiscale Discontinuous Galerkin Method (GMsDGM). In this method, we construct a multiscale space using solution of the local spectral problems in each coarse elements. The results of the numerical solution for the two-dimensional problem are presented for model problems of the wave propagation in fractured media.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. De Basabe, J.D., Sen, M.K., Wheeler, M.F.: Seismic wave propagation in fractured media: a discontinuous Galerkin approach, SEG Expanded Abstr 30 (2011)

    Google Scholar 

  2. De Basabe, J.D., Sen, M.K., Wheeler, M.F.: Elastic wave propagation in fractured media using the discontinuous Galerkin method. Geophysics 81(4), T163–T174 (2016)

    Article  Google Scholar 

  3. Zhang, J.: Elastic wave modeling in fractured media with an explicit approach. Geophysics 70(5), T75–T85 (2005)

    Article  Google Scholar 

  4. Schoenberg, M.: Elastic wave behavior across linear slip interfaces. J. Acoust. Soc. Am. 68(5), 1516–1521 (1980)

    Article  Google Scholar 

  5. Engquist, B., Majda, A.: Absorbing boundary conditions for numerical simulation of waves. Proc. Nat. Acad. Sci. 74(5), 1765–1766 (1977)

    Article  MathSciNet  Google Scholar 

  6. Grote, M.J., Schneebeli, A., Schötzau, D.: Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44(6), 2408–2430 (2006)

    Article  MathSciNet  Google Scholar 

  7. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)

    Article  MathSciNet  Google Scholar 

  8. Lahivaara, T.: Discontinuous Galerkin Method for Time-Domain Wave Problems. University of Eastern Finland, Joensuu (2010)

    MATH  Google Scholar 

  9. Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 215, 116–135 (2013)

    Article  MathSciNet  Google Scholar 

  10. Chung, E.T., Efendiev, Y., Leung, W.T.: An online generalized multiscale discontinuous Galerkin method (GMsDGM) for flows in heterogeneous media. Commun. Comput. Phys. 21(2), 401–422 (2017)

    Article  MathSciNet  Google Scholar 

  11. Chung, E.T., Efendiev, Y., Leung, W.T.: Generalized multiscale finite element methods for wave propagation in heterogeneous media. Multiscale Model. Simul. 12(4), 1641–1721 (2014)

    Article  MathSciNet  Google Scholar 

  12. Gao, K., Fu, S., Gibson, R.L., Chung, E.T., Efendiev, Y.: Generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media. J. Comput. Phys. 295, 161–188 (2015)

    Article  MathSciNet  Google Scholar 

  13. Chung, E.T., Efendiev, Y., Gibson, R.L., Vasilyeva, M.: A generalized multiscale finite element method for elastic wave propagation in fractured media. GEM-Int. J. Geomath. 7(2), 163–182 (2016)

    Article  MathSciNet  Google Scholar 

  14. Chung, E.T., Efendiev, Y., Fu, S.: Generalized multiscale finite element method for elasticity equations. GEM-Int. J. Geomath. 5(2), 225–251 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

Work is supported by the mega-grant of the Russian Federation Government (N 14.Y26.31.0013).

Author information

Authors and Affiliations

  1. Multiscale Model Reduction Laboratory, North-Eastern Federal University, Yakutsk, Russia

    U. Gavrileva, V. Alekseev & M. Vasilyeva

  2. Institute for Scientific Computation, Texas A&M University, College Station, TX, USA

    M. Vasilyeva

  3. Department of Seismology, Earth Sciences Division, CICESE, Baja California, Mexico

    J. D. De Basabe

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

    Y. Efendiev

  5. Department of Geology and Geophysics, Texas A&M University, College Station, TX, USA

    R. L. Gibson Jr.

Authors
  1. U. Gavrileva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Alekseev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Vasilyeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. D. De Basabe
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Y. Efendiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. R. L. Gibson Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to U. Gavrileva .

Editor information

Editors and Affiliations

  1. IICT, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. University of Rousse, Rousse, Bulgaria

    Lubin Vulkov

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gavrileva, U., Alekseev, V., Vasilyeva, M., De Basabe, J.D., Efendiev, Y., Gibson, R.L. (2019). Generalized Multiscale Discontinuous Galerkin Method for Helmholtz Problem in Fractured Media. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Finite Difference Methods. Theory and Applications. FDM 2018. Lecture Notes in Computer Science(), vol 11386. Springer, Cham. https://doi.org/10.1007/978-3-030-11539-5_27

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-11539-5_27

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-11538-8

  • Online ISBN: 978-3-030-11539-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation