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

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Numerical Mathematics and Advanced Applications ENUMATH 2017 (ENUMATH 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 126))

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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.

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References

  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)

    Article  MathSciNet  Google Scholar 

  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. M. Bause, Iterative coupling of mixed and discontinuous Galerkin methods for poroelasticity. arXiv:1802.03230 (2018)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  7. M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12(2), 155–164 (1941)

    Article  Google Scholar 

  8. M. Borregales, J.M. Nordbotten, K. Kumar, F.A. Radu, Robust iterative schemes for non-linear poromechanics. Comput. Geosci. 22, 1021–1038 (2018)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  13. U. Köcher, Space-time-parallel poroelasticity simulation. arXiv:1801.04984 (2018)

    Google Scholar 

  14. F. List, F.A. Radu, A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20(2), 341–353 (2016)

    Article  MathSciNet  Google Scholar 

  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. A. Mikelić, M.F. Wheeler, Convergence of iterative coupling for coupled flow and geomechanics. Comput. Geosci. 18(3–4), 325–341 (2013)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics, University of Bergen, Bergen, Norway

    Manuel Borregales & Florin Adrian Radu

Authors
  1. Manuel Borregales
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  2. Florin Adrian Radu
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Corresponding author

Correspondence to Manuel Borregales .

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Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Florin Adrian Radu

  2. University of Bergen, Bergen, Norway

    Kundan Kumar

  3. University of Bergen, Bergen, Norway

    Inga Berre

  4. University of Bergen, Bergen, Norway

    Jan Martin Nordbotten

  5. Hasselt University, Diepenbeek, Belgium

    Iuliu Sorin Pop

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Borregales, M., Radu, F.A. (2019). Higher Order Space-Time Elements for a Non-linear Biot Model. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_49

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