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Higher Order Space-Time Elements for a Non-linear Biot Model

  • Manuel BorregalesEmail author
  • Florin Adrian Radu
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

In this work, we consider a non-linear extension of the linear, quasi-static Biot’s model. Precisely, we assume that the volumetric strain and the fluid compressibility are non-linear functions. We propose a fully discrete numerical scheme for this model based on higher order space-time elements. We use mixed finite elements for the flow equation, (continuous) Galerkin finite elements for the mechanics and discontinuous Galerkin for the time discretization. We further use the L-scheme for linearising the system appearing on each time step. The stability of this approach is illustrated by a numerical experiment.

Notes

Acknowledgements

This work was partially supported by the NFR-DAAD project EDIFY 255715 and the NFR project SUCCESS.

References

  1. 1.
    T. Almani, K. Kumar, A.H. Dogru, G. Singh, M.F. Wheeler, Convergence analysis of multirate fixed-stress split iterative schemes for coupling flow with geomechanics. Comput. Methods. Appl. Mech. Eng. 311, 180–207 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    W. Bangerth, G. Kanschat, T. Heister, L. Heltai, G. Kanschat, The deal.II library version 8.4. J. Numer. Math. 24, 135–141 (2016)Google Scholar
  3. 3.
    M. Bause, Iterative coupling of mixed and discontinuous Galerkin methods for poroelasticity. arXiv:1802.03230 (2018)Google Scholar
  4. 4.
    M. Bause, U. Köcher, Variational time discretization for mixed finite element approximations of nonstationary diffusion problems. J. Comput. Appl. Math. 289, 208–224 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Bause, F. Radu, U. Köcher, Space–time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Methods. Appl. Mech. Eng. 320, 745–768 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Bause, F.A. Radu, U. Köcher, Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space. Numer. Math. 137(4), 773–818 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)CrossRefGoogle Scholar
  8. 8.
    M. Borregales, J.M. Nordbotten, K. Kumar, F.A. Radu, Robust iterative schemes for non-linear poromechanics. Comput. Geosci. 22, 1021–1038 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. Borregales, K. Kumar, F.A. Radu, C. Rodrigo, F.J. Gaspar, A parallel-in-time fixed-stress splitting method for Biot’s consolidation model. arXiv:1802.00949 (2018)Google Scholar
  10. 10.
    J.W. Both, M. Borregales, J.M. Nordbotten, K. Kumar, F.A. Radu, Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    F.J. Gaspar, C. Rodrigo, On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics. Comput. Methods Appl. Mech. Eng. 326, 526–540 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    J. Kim, H. Tchelepi, R. Juanes, Stability and convergence of sequential methods for coupled flow and geomechanics: fixed-stress and fixed-strain splits. Comput. Methods Appl. Mech. Eng. 200(13–16), 1591–1606 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    U. Köcher, Space-time-parallel poroelasticity simulation. arXiv:1801.04984 (2018)Google Scholar
  14. 14.
    F. List, F.A. Radu, A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20(2), 341–353 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Mikelić, M.F. Wheeler, Theory of the dynamic Biot-Allard equations and their link to the quasi-static Biot system. J. Math. Phys. 53(12), 123702 (2012)Google Scholar
  16. 16.
    A. Mikelić, M.F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 18(3–4), 325–341 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    I. Pop, F. Radu, P. Knabner, Mixed finite elements for the Richards’ equation: linearization procedure. J. Comput. Appl. Math. 168(1–2), 365–373 (2004)MathSciNetCrossRefGoogle Scholar
  18. 18.
    F.A. Radu, J.M. Nordbotten, I.S. Pop, K. Kumar, A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. 289, 134–141 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    C. Rodrigo, X. Hu, P. Ohm, J.H. Adler, F.J. Gaspar, L. Zikatanov, New stabilized discretizations for poroelasticity and the Stokes’ equations. arXiv:1706.05169 (2017)Google Scholar
  20. 20.
    J.A. White, N. Castelletto, H.A. Tchelepi, Block-partitioned solvers for coupled poromechanics: a unified framework. Comput. Methods. Appl. Mech. Eng. 303, 55–74 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway

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