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Published in:
Control and Nash Games with Mean Field Effect Read chapter Read first chapter

2014 | OriginalPaper | Chapter

The Rain on Underground Porous Media

Part I: Analysis of a Richards Model

Authors : Christine Bernardi, Adel Blouza, Linda El Alaoui

Published in: Partial Differential Equations: Theory, Control and Approximation

Publisher: Springer Berlin Heidelberg

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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.

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previous chapter Control and Nash Games with Mean Field Effect
next chapter Finite Volume Multilevel Approximation of the Shallow Water Equations
Literature
1.
2.
3.
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
4.
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)
5.
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) MathSciNetCrossRef
6.
Fabrié, P., Gallouët, T.: Modelling wells in porous media flows. Math. Models Methods Appl. Sci. 10, 673–709 (2000) MathSciNetMATH
7.
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)
8.
Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations. Lecture Notes in Mathematics, vol. 749. Springer, Berlin (1979) CrossRefMATH
9.
Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer, Berlin (1986) CrossRefMATH
10.
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)
11.
Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. I. Dunod, Paris (1968) MATH
12.
Nochetto, R.H., Verdi, C.: Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal. 25, 784–814 (1988) MathSciNetCrossRefMATH
13.
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) MathSciNetCrossRefMATH
14.
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) MathSciNetCrossRefMATH
15.
Richards, L.A.: Capillary conduction of liquids through porous mediums. Physics 1, 318–333 (1931) CrossRefMATH
16.
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) MathSciNetCrossRefMATH
17.
Sochala, P., Ern, A.: Numerical methods for subsurface flows and coupling with runoff. (2013, to appear)
18.
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) MathSciNetCrossRefMATH
19.
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) MathSciNetCrossRefMATH
Metadata
Title
The Rain on Underground Porous Media
Authors
Christine Bernardi
Adel Blouza
Linda El Alaoui
Copyright Year
2014
Publisher
Springer Berlin Heidelberg
Book
Partial Differential Equations: Theory, Control and Approximation

Print ISBN: 978-3-642-41400-8

Electronic ISBN: 978-3-642-41401-5

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-642-41401-5_2

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