Abstract
The Richards equation models the water flow in a partially saturated underground porous medium under the surface. When it rains on the surface, boundary conditions of Signorini type must be considered on this part of the boundary. The authors first study this problem which results into a variational inequality and then propose a discretization by an implicit Euler’s scheme in time and finite elements in space. The convergence of this discretization leads to the well-posedness of the problem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311–341 (1983)
Alt, H.W., Luckhaus, S., Visintin, A.: On nonstationary flow through porous media. Ann. Mat. Pura Appl. 136, 303–316 (1984)
Bernardi, C., El Alaoui, L., Mghazli, Z.: A posteriori analysis of a space and time discretization of a nonlinear model for the flow in variably saturated porous media. Submitted
Berninger, H.: Domain decomposition methods for elliptic problems with jumping nonlinearities and application to the Richards equation. Ph.D. Thesis, Freie Universität, Berlin, Germany (2007)
Brezzi, F., Hager, W.W., Raviart, P.A.: Error estimates for the finite element solution to variational inequalities. II. Mixed methods. Numer. Math. 31, 1–16 (1978/1979)
Fabrié, P., Gallouët, T.: Modelling wells in porous media flows. Math. Models Methods Appl. Sci. 10, 673–709 (2000)
Gabbouhy, M.: Analyse mathématique et simulation numérique des phénomènes d’écoulement et de transport en milieux poreux non saturés. Application à la région du Gharb. Ph.D. Thesis, University Ibn Tofail, Kénitra, Morocco (2000)
Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics, vol. 749. Springer, Berlin (1979)
Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer, Berlin (1986)
Glowinski, R., Lions, J.L., Trémolières, R.: Analyse numérique des inéquations variationnelles. 2. Applications aux phénomènes stationnaires et d’évolution, Collection. Méthodes Mathématiques de l’Informatique, vol. 5. Dunod, Paris (1976)
Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. I. Dunod, Paris (1968)
Nochetto, R.H., Verdi, C.: Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal. 25, 784–814 (1988)
Radu, F., Pop, I.S., Knabner, P.: Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation. SIAM J. Numer. Anal. 42, 1452–1478 (2004)
Rajagopal, K.R.: On a hierarchy of approximate models for flows of incompressible fluids through porous solid. Math. Models Methods Appl. Sci. 17, 215–252 (2007)
Richards, L.A.: Capillary conduction of liquids through porous mediums. Physics 1, 318–333 (1931)
Schneid, E., Knabner, P., Radu, F.: A priori error estimates for a mixed finite element discretization of the Richards’ equation. Numer. Math. 98, 353–370 (2004)
Sochala, P., Ern, A.: Numerical methods for subsurface flows and coupling with runoff. (2013, to appear)
Sochala, P., Ern, A., Piperno, S.: Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows. Comput. Methods Appl. Mech. Eng. 198, 2122–2136 (2009)
Woodward, C.S., Dawson, C.N.: Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media. SIAM J. Numer. Anal. 37, 701–724 (2000)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bernardi, C., Blouza, A., El Alaoui, L. (2014). The Rain on Underground Porous Media. In: Ciarlet, P., Li, T., Maday, Y. (eds) Partial Differential Equations: Theory, Control and Approximation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41401-5_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-41401-5_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-41400-8
Online ISBN: 978-3-642-41401-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)