Skip to main content
Springer Nature Link
Log in

Iterative Coupling of Variational Space-Time Methods for Biot’s System of Poroelasticity

  • Conference paper
  • First Online:
Numerical Mathematics and Advanced Applications ENUMATH 2015

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 112))

  • 1579 Accesses

Abstract

In this work we present an iterative coupling scheme for the quasi-static Biot system of poroelasticity. For the discretization of the subproblems describing mechanical deformation and single-phase flow space-time finite element methods based on a discontinuous Galerkin approximation of the time variable are used. The spatial approximation of the flow problem is done by mixed finite element methods. The stability of the approach is illustrated by numerical experiments. The presented variational space-time framework is of higher order accuracy such that problems with high fluctuations become feasible. Moreover, it offers promising potential for the simulation of the fully dynamic Biot–Allard system coupling an elastic wave equation for solid’s deformation with single-phase flow for fluid infiltration.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Bangerth, R. Rannacher, Adaptive Methods for Differential Equations (Birkhäuser, Basel, 2003)

    Book  MATH  Google Scholar 

  2. M. Bause, U. Köcher, Variational time discretization for mixed finite element approximations of nonstationary diffusion problems. J. Comput. Appl. Math. 289, 208–224 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bause, F. Radu, U. Köcher, Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space. Numer. Math. 1–42 (2015, subm.). http://arxiv.org/abs/1504.04491

  4. Z. Chen, Finite Element Methods and Their Applications (Springer, Berlin, 2010)

    Google Scholar 

  5. A. Ern, F. Schieweck, Discontinuous Galerkin method in time combined with an stabilized finite element method in space for linear first-order PDEs. Math. Comput. 1–33 (2014). Electronically published on January 11, 2016, http://dx.doi.org/10.1090/mcom/3073, http://hal.archives-ouvertes.fr/hal-00947695

  6. S. Hussain, F. Schieweck, S. Turek, Higher order Galerkin time discretization for nonstationary incompressible flow, in Numerical Mathematics and Advanced Applications 2011, ed. by A. Cangiani et al. (Springer, Berlin, 2013), pp. 509–517

    Chapter  Google Scholar 

  7. B. Jha, R. Juanes, A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics. Acta Geotechnica 2, 139–153 (2007)

    Article  Google Scholar 

  8. O. Karakashin, C. Makridakis, Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations. Math. Comput. Am. Math. Soc. 74, 85–102 (2004)

    Article  MathSciNet  Google Scholar 

  9. J. Kim, H.A. Tchelepi, R. Juanes, Stability and convergence of sequential methods for coupled flow and geomechanics: drained and undrained splits. Comput. Methods Appl. Mech. Eng. 200, 2094–2116 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. U. Köcher, Variational space-time methods for the elastic wave equation and the diffusion equation, PhD thesis, Helmut-Schmidt-Universität (2015). http://edoc.sub.uni-hamburg.de/hsu/volltexte/2015/3112/

  11. U. Köcher, M. Bause, Variational space-time methods for the wave equation. J. Sci. Comput. 61, 424–453 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Mikelić, M. Wheeler, Theory of the dynamic Biot–Allard equations and their link to the quasi-static Biot system. J. Math. Phys. 53, 123702:1–15 (2012)

    Google Scholar 

  13. A. Mikelić, M. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17, 479–496 (2013)

    Article  MathSciNet  Google Scholar 

  14. A. Mikelić et al., Numerical convergence study of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 18, 325–341 (2014)

    Article  MathSciNet  Google Scholar 

  15. P.J. Philips, M. Wheeler, A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I, II. Comput. Geosci. 11, 131–158 (2007)

    Article  Google Scholar 

  16. A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations (Springer, Berlin, 2008)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Helmut Schmidt University, Holstenhofweg 85, 22043, Hamburg, Germany

    Markus Bause & Uwe Köcher

Authors
  1. Markus Bause
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uwe Köcher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus Bause .

Editor information

Editors and Affiliations

  1. Mathematics & Applied Mathematics, Middle East Technical University Mathematics & Applied Mathematics, Ankara, Turkey

    Bülent Karasözen

  2. Dept of Computer Engg, Middle East Technical Univ Dept of Computer Engg, Cankaya, Ankara, Turkey

    Murat Manguoğlu

  3. Middle East Technical University Mathematics & Institute of Applied Mathe, Ankara, Turkey

    Münevver Tezer-Sezgin

  4. Civil Engineering & Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Serdar Göktepe

  5. Institute of Applied Mathematics, Middle East Technical University Institute of Applied Mathematics, Ankara, Turkey

    Ömür Uğur

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Bause, M., Köcher, U. (2016). Iterative Coupling of Variational Space-Time Methods for Biot’s System of Poroelasticity. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_15

Download citation

Publish with us

Policies and ethics