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A Linear Domain Decomposition Method for Two-Phase Flow in Porous Media

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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

This article is a follow up of our submitted paper (D. Seus et al, Comput Methods Appl Mech Eng 333:331–355, 2018) in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of two-phase in porous media and the convergence of the proposed domain decomposition method is rigorously shown.

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Notes

  1. 1.

    Similar assumptions are used in the literature, cf. [10]. More recently, the case of Hölder continuity has been treated, see [12].

References

  1. E. Ahmed, S. Ali Hassan, C. Japhet, M. Kern, M. Vohralík, A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types (2017). https://hal.inria.fr/hal-01540956

  2. H. Berninger, O. Sander, Substructuring of a Signorini-type problem and Robin’s method for the Richards equation in heterogeneous soil. Comput. Vis. Sci. 13(5), 187–205 (2010)

    Article  MathSciNet  Google Scholar 

  3. H. Berninger, R. Kornhuber, O. Sander, A multidomain discretization of the Richards equation in layered soil. Comput. Geosci. 19(1), 213–232 (2015)

    Article  MathSciNet  Google Scholar 

  4. M. Borregales, F.A. Radu, K. Kumar, J.M. Nordbotten, Robust iterative schemes for non-linear poromechanics. Comput. Geosci. 22(4), 1021–1038 (2018)

    Article  MathSciNet  Google Scholar 

  5. J. Both, M. Borregales, J. Nordbotten, K. Kumar, F. Radu, Robust fixed stress splitting for Biot’s equations in heterogeneous media. Appl. Math. Lett. 68, 101–108 (2017)

    Article  MathSciNet  Google Scholar 

  6. R. Helmig, Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems (Springer, Berlin, 1997)

    Book  Google Scholar 

  7. S. Karpinski, I.S. Pop, F.A. Radu, Analysis of a linearization scheme for an interior penalty discontinuous Galerkin method for two-phase flow in porous media with dynamic capillarity effects. Int. J. Numer. Methods Eng. 112(6), 553–577 (2017)

    Article  MathSciNet  Google Scholar 

  8. P.-L. Lions, On the Schwarz alternating method, in Proceedings of the 1st International Symposium on Domain Decomposition Methods for Partial Differential Equations, ed. by R. Glowinski, G.H. Golub, G.A. Meurant, J. Periaux (SIAM, Philadelphia, 1988), pp. 1–42

    Google Scholar 

  9. 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 

  10. I.S. Pop, F.A. 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 

  11. 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 

  12. F.A. Radu, K. Kumar, J.M. Nordbotten, I.S. Pop, A robust, mass conservative scheme for two-phase flow in porous media including Hölder continuous nonlinearities. IMA J. Numer. Anal. 38, 884–920 (2017)

    Article  Google Scholar 

  13. D. Seus, K. Mitra, I.S. Pop, F.A. Radu, C. Rohde, A linear domain decomposition method for partially saturated flow in porous media. Comput. Methods Appl. Mech. Eng. 333, 331–355 (2018)

    Article  MathSciNet  Google Scholar 

  14. M. Slodička, A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media. SIAM J. Sci. Comput. 23(5), 1593–1614 (2002)

    Article  MathSciNet  Google Scholar 

  15. I. Yotov, A mixed finite element discretization on non–matching multiblock grids for a degenerate parabolic equation arizing in porous media flow. East-West J. Numer. Math. 5, 211–230 (1997)

    MathSciNet  MATH  Google Scholar 

  16. I. Yotov, M.F. Wheeler, Domain decomposition mixed methods for multiphase groundwater flow in multiblock aquifers, in Computer Methods in Water Resources XII. Transactions on Ecology and the Environment, vol. 17 (Wit Press, Ashurst, 1998), pp. 59–66

    Google Scholar 

  17. I. Yotov, Scientific computing and applications, ed. by P. Minev, Y. Lin (Nova Science Publishers, Commack, 2001), pp. 157–167

    Google Scholar 

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Acknowledgements

This work was partially supported by the NFR supported project CHI #25510 and by the VISTA project #6367.

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Authors and Affiliations

  1. University of Stuttgart, Institute of Applied Analysis and Numerical Simulation, Stuttgart, Germany

    David Seus & Christian Rohde

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

    Florin A. Radu

Authors
  1. David Seus
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  2. Florin A. Radu
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  3. Christian Rohde
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Corresponding author

Correspondence to David Seus .

Editor information

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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Seus, D., Radu, F.A., Rohde, C. (2019). A Linear Domain Decomposition Method for Two-Phase Flow in Porous Media. 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_55

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