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Substructuring of a Signorini-type problem and Robin’s method for the Richards equation in heterogeneous soil

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Computing and Visualization in Science

An Erratum to this article was published on 01 June 2011

Abstract

We prove a substructuring result for variational inequalities. It concerns but is not restricted to the Richards equation in heterogeneous soil, and it includes boundary conditions of Signorini’s type. This generalizes existing results for the linear case and leads to interface conditions known from linear variational equalities: continuity of Dirichlet and flux values in a weak sense. In case of the Richards equation, these are the continuity of the physical pressure and of the water flux, which is hydrologically reasonable. We use these interface conditions in a heterogeneous problem with piecewise constant soil parameters, which we address by the Robin method. We prove that, for a certain time discretization, the homogeneous problems in the subdomains including Robin and Signorini-type boundary conditions can be solved by convex minimization. As a consequence, we are able to apply monotone multigrid in the discrete setting as an efficient and robust solver for the local problems. Numerical results demonstrate the applicability of our approach.

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References

  1. Bastian P., Ippisch O., Rezanezhad F., Vogel H.J., Roth K.: Numerical simulation and experimental studies of unsaturated water flow in heterogeneous systems. In: Jäger, W., Rannacher, R., Warnatz, J. (eds) Reactive Flows, Diffusion and Transport, pp. 579–598. Springer, Berlin (2005)

    Google Scholar 

  2. Bastian P., Birken K., Johannsen K., Lang S., Neuß N., Rentz-Reichert H., Wieners C.: UG—A flexible software toolbox for solving partial differential equations. Comput. Vis. Sci. 1(1), 27–40 (1997)

    Article  MATH  Google Scholar 

  3. Bastian P., Blatt M., Dedner A., Engwer C., Klöfkorn R., Kornhuber R., Ohlberger M., Sander O.: A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in dune. Computing 82((2–3), 121–138 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bear J.: Dynamics of Fluids in Porous Media. Dover Publications, NY (1988)

    MATH  Google Scholar 

  5. Berninger H.: Non-overlapping domain decomposition for the Richards equation via superposition operators. In: Bercovier, M., Gander, M.J., Kornhuber, R., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XVIII, volume 70 of LNCSE, pp. 169–176. Springer, Berlin (2009)

    Chapter  Google Scholar 

  6. Berninger, H.: Domain Decomposition Methods for Elliptic Problems with Jumping Nonlinearities and Application to the Richards Equation. PhD thesis, Freie Universität Berlin, (2007)

  7. Berninger, H., Discacciati M.: Analysis of the Robin method for nonlinear transmission problems. In preparation

  8. Berninger H., Kornhuber R., Sander O.: On nonlinear Dirichlet–Neumann algorithms for jumping nonlinearities. In: Widlund, O.B., Keyes, D.E. (eds) Domain Decomposition Methods in Science and Engineering XVI, volume 55 of LNCSE, pp. 483–490. Springer, Berlin (2007)

    Google Scholar 

  9. Berninger, H., Kornhuber R., Sander O.: Convergence behaviour of Dirichlet–Neumann and Robin methods for a nonlinear transmission problem. In: Domain Decomposition Methods in Science and Engineering XIX, LNCSE. Springer, (2010), to appear

  10. Berninger, H., Kornhuber, R., Sander, O.: Solution of the Richards equation in heterogeneous soil. In preparation

  11. Berninger, H., Kornhuber, R., Sander, O.: Fast and robust numerical solution of the Richards equation in homogeneous soil. Technical report, Freie Universität Berlin. J. Numer. Anal. Preprint A /01/2010. submitted to SIAM

  12. Brezzi F., Gilardi G.: Functional spaces. In: Kardestuncer, H., Norrie, D.H. (eds) Finite Element Handbook, chapter 2 (part 1), pp. 1.29–1.75. Springer, Berlin (1987)

    Google Scholar 

  13. Brooks, R.J., Corey, A.T.: Hydraulic properties of porous media. Technical Report Hydrology Paper No. 3, Colorado State University, Civil Engineering Department, Fort Collins, (1964)

  14. Burdine, N.T.: Relative permeability calculations from pore-size distribution data. Petr. Trans., Am. Inst. Mining Metall. Eng. (198), 71–77 (1953)

  15. Discacciati, M.: Domain decomposition methods for the coupling of surface and groundwater flows. PhD thesis, Ecole Polytechnique Fédérale de Lausanne (2004)

  16. Dubois, O.: Optimized Schwarz methods for the advection-diffusion equation and for problems with discontinuous coefficients. PhD thesis (2007)

  17. Ekeland, I., Temam, R.: Convex analysis and variational problems. North–Holland (1976)

  18. Fuhrmann J.: Zur Verwendung von Mehrgitterverfahren bei der numerischen Behandlung elliptischer partieller Differentialgleichungen mit variablen Koeffizienten. PhD thesis, TU Chemnitz–Zwickau (1994)

  19. Fuhrmann J., Langmach H.: Stability and existence of solutions of time-implicit finite volume schemes for viscous nonlinear conservation laws. Appl. Numer. Math. 37(1–2), 201–230 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gander M.J.: Optimized Schwarz methods. SIAM J. Numer. Anal. 44(2), 699–731 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Helmig R., Weiss A., Wohlmuth B.: Dynamic capillary effects in heterogeneous porous media. Comput. Geosci. 11, 261–274 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Helmig, R., Weiss, A., Wohlmuth, B.I.: Variational inequalities for modeling flow in heterogeneous porous media with entry pressure. Comput. Geosci. (3), 373–389 (2009)

  23. Kikuchi N., Oden J.T.: Contact Problems In Elasticity: A Study of Variational Inequalities And Finite Element Methods, volume 8. SIAM Studies in Applied Mathematics, Thailand (1988)

    Google Scholar 

  24. Kornhuber R.: On constrained Newton linearization and multigrid for variational inequalities. Numer. Math. 91, 699–721 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kornhuber, R.: Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems Teubner (1997)

  26. Leoni G., Morini M.: Necessary and sufficient conditions for the chain rule in \({W_{{\rm loc}}^{1,1}(\mathbb{R}^N;\mathbb{R}^d)}\) and \({BV_{{\rm loc}}(\mathbb{R}^N;\mathbb{R}^d)}\) . J. Eur. Math. Soc. (JEMS) 9(2), 219–252 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lions J.-L., Magenes E.: Non-Homogeneous Boundary Value Problems and Applications vol. 1. Springer, Berlin (1972)

    Google Scholar 

  28. Lui S.H.: A Lions non-overlapping domain decomposition method for domains with an arbitrary interface. IMA J. Numer. Anal. 29(2), 332–349 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Marcus M., Mizel V.J.: Complete characterization of functions which act, via superposition, on Sobolev spaces. Trans. Am. Math. Soc. 251, 187–218 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  30. Marcus M., Mizel V.J.: Every superposition operator mapping one Sobolev space into another is continuous. J. Funct. Anal. 33, 217–229 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  31. Quarteroni A., Valli A.: Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, Oxford (1999)

    MATH  Google Scholar 

  32. Rawls W.J., Ahuja L.R., Brakensiek D.L., Shirmohammadi A.: Infiltration and soil water movement. In: Maidment, D.R. (eds) Handbook of Hydrology, chapter 5, McGraw–Hill, New York (1993)

    Google Scholar 

  33. Richards L.A.: Capillary conduction of liquids through porous mediums. Physics 1, 318–333 (1931)

    Article  MATH  Google Scholar 

  34. Stalling D., Westerhoff M., Hege H.-C.: Amira: A highly interactive system for visual data analysis. In: Hansen, C., Johnson, C. (eds) The Visualization Handbook, Chap. 38, pp. 749–767. Elsevier, Amterdam (2005)

    Chapter  Google Scholar 

  35. van Genuchten M.T.: A closed–form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44, 892–898 (1980)

    Article  Google Scholar 

  36. Werner D.: Funktionalanalysis, 5th edn. Springer, Berlin (2005)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195, Berlin, Germany

    Heiko Berninger & Oliver Sander

Authors
  1. Heiko Berninger
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  2. Oliver Sander
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Corresponding author

Correspondence to Heiko Berninger.

Additional information

Communicated by Gabriel Wittum.

This work was supported by the BMBF–Förderprogramm “Mathematik für Innovationen in Industrie und Dienstleistungen”.

An erratum to this article can be found at http://dx.doi.org/10.1007/s00791-012-0173-0

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Berninger, H., Sander, O. Substructuring of a Signorini-type problem and Robin’s method for the Richards equation in heterogeneous soil. Comput. Visual Sci. 13, 187–205 (2010). https://doi.org/10.1007/s00791-010-0141-5

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  • DOI: https://doi.org/10.1007/s00791-010-0141-5

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