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

  • Markus Bause
  • Uwe Köcher
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 112)

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.

Keywords

Mixed Finite Element Finite Element Space Mixed Finite Element Method Fixed Point Iteration Time Discretization Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    W. Bangerth, R. Rannacher, Adaptive Methods for Differential Equations (Birkhäuser, Basel, 2003)CrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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. 4.
    Z. Chen, Finite Element Methods and Their Applications (Springer, Berlin, 2010)Google Scholar
  5. 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. 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. 509–517CrossRefGoogle Scholar
  7. 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)CrossRefGoogle Scholar
  8. 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)MathSciNetCrossRefGoogle Scholar
  9. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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. 11.
    U. Köcher, M. Bause, Variational space-time methods for the wave equation. J. Sci. Comput. 61, 424–453 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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. 13.
    A. Mikelić, M. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 17, 479–496 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Mikelić et al., Numerical convergence study of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 18, 325–341 (2014)MathSciNetCrossRefGoogle Scholar
  15. 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)CrossRefGoogle Scholar
  16. 16.
    A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations (Springer, Berlin, 2008)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Helmut Schmidt UniversityHamburgGermany

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