Show Summary Details
More options …

# Advances in Pure and Applied Mathematics

Editor-in-Chief: Trimeche, Khalifa

Editorial Board: Aldroubi, Akram / Anker, Jean-Philippe / Aouadi, Saloua / Bahouri, Hajer / Baklouti, Ali / Bakry, Dominique / Baraket, Sami / Ben Abdelghani, Leila / Begehr, Heinrich / Beznea, Lucian / Bezzarga, Mounir / Bonami, Aline / Demailly, Jean-Pierre / Fleckinger, Jacqueline / Gallardo, Leonard / Ismail, Mourad / Jarboui, Noomen / Jouini, Elyes / Karoui, Abderrazek / Kamoun, Lotfi / Kobayashi, Toshiyuki / Maday, Yvon / Marzougui, Habib / Mili, Maher / Mustapha, Sami / Ovsienko, Valentin / Peigné, Marc / Pouzet, Maurice / Radulescu, Vicentiu / Schwartz, Lionel / Sifi, Mohamed / Zaag, Hatem / Zarati, Said

CiteScore 2017: 1.29

SCImago Journal Rank (SJR) 2017: 0.177
Source Normalized Impact per Paper (SNIP) 2017: 0.409

Mathematical Citation Quotient (MCQ) 2017: 0.21

Online
ISSN
1869-6090
See all formats and pricing
More options …
Volume 9, Issue 1

# A priori error analysis of the implicit Euler, spectral discretization of a nonlinear equation for a flow in a partially saturated porous media

Nahla Abdellatif
• Corresponding author
• University of Manouba, ENSI, Campus Universitaire de Manouba, 2010 Manouba, Tunisia; and University of Tunis El Manar, ENIT, LAMSIN, BP 37, Le Belvédère, 1002 Tunis, Tunisia
• Email
• Other articles by this author:
/ Christine Bernardi
• Laboratoire Jacques-Louis Lions, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France
• Email
• Other articles by this author:
/ Moncef Touihri
/ Driss Yakoubi
Published Online: 2017-05-11 | DOI: https://doi.org/10.1515/apam-2016-0084

## Abstract

The aim of this work is the numerical study of a nonlinear equation, which models the water flow in a partially saturated underground porous medium under the surface. We propose a discretization of this equation that combines Euler’s implicit scheme in time and spectral methods in space. We prove optimal error estimates between the continuous and discrete solutions. Some numerical experiments confirm the interest of this approach. We present numerical experiments which are in perfect coherence with the analysis.

MSC 2010: 76D05; 76S05

## References

• [1]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), 311–341.

• [2]

T. Arbogast, An error analysis for Galerkin approximations to an equation of mixed elliptic-parabolic type, Technical Report TR90-33, Department of Computational and Applied Mathematics, Rice University, Houston, 1990. Google Scholar

• [3]

T. Arbogast, M. Obeyesekere and M. F. Wheeler, Numerical methods for the simulation of flow in root-soil systems, SIAM J. Numer. Anal. 30 (1993), 1677–1702.

• [4]

T. Arbogast, M. F. Wheeler and N. Y. Zhang, A non-linear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Numer. Anal. 33 (1996), 1669–1687.

• [5]

M. Azaïez, F. Ben Belgacem, M. Grundmann and H. Khallouf, Staggered grids hybrid-dual spectral element method for second order elliptic problems. Application to high-order time splitting for Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 166 (1998), 183–199.

• [6]

J. W. Barrett and P. Knabner, Finite element approximation of the transport of reactive solutes in porous media. Part II: Error estimates for equilibrium adsorption processes, SIAM J. Numer. Anal. 34 (1997), 455–479.

• [7]

J.-M. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary, Chin. Ann. Math. Ser. B 32 (2011), 823–846.

• [8]

C. Bernardi, L. El Alaoui and Z. Mghazli, A posteriori analysis of a space and time discretization of a nonlinear model for the flow in partially saturated porous media, IMA J. Numer. Anal. 34 (2014), 1002–1036.

• [9]

C. Bernardi and Y. Maday, Spectral methods, Handbook of Numerical Analysis. Vol. 5, North-Holland, Amsterdam (1997), 209–485. Google Scholar

• [10]

C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques, Math. Appl. (Berlin) 45, Springer, Berlin, 2004. Google Scholar

• [11]

H. Brezis and P. Mironescu, Gagliardo–Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ. 1 (2001), 387–404.

• [12]

F. Brezzi, J. Rappaz and P.-A. Raviart, Finite dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions, Numer. Math. 36 (1980), 1–25.

• [13]

S. Del Pino and O. Pironneau, A fictitious domain based on general pde’s solvers, Numerical Methods for Scientific Computing Variational Problems and Applications, CIMNE, Barcelona (2003), 1–11. Google Scholar

• [14]

R. Eymard, M. Gutnic and D. Hilhorst, The finite volume method for Richards equation, Comput. Geosci. 3 (1999), 259–294.

• [15]

M. Gabbouhy, 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, Université Ibn Tofail, Kénitra, 2000. Google Scholar

• [16]

S. M. F. Garcia, Improved error estimates for mixed finite-element approximations for nonlinear parabolic equations: The continuous-time case, Numer. Methods Partial Differential Equations 10 (1994), 129–147.

• [17]

S. M. F. Garcia, Improved error estimates for mixed finite-element approximations for nonlinear parabolic equations: The discrete-time case, Numer. Methods Partial Differential Equations 10 (1994), 149–169.

• [18]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms, Springer, Berlin, 1986. Google Scholar

• [19]

J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod & Gauthier–Villars, Paris, 1969. Google Scholar

• [20]

J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. I, Dunod, Paris, 1968. Google Scholar

• [21]

F. List and F. Radu, A study on iterative methods for Richards’ equation, Comput. Geosci. 20 (2016), no. 2, 341–353.

• [22]

Y. Maday and E. M. Rønquist, Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries, Comput. Methods Appl. Mech. Engrg. 80 (1990), 91–115.

• [23]

J.-C. Nédélec, Mixed finite elements in ${ℝ}^{3}$, Numer. Math. 35 (1980), 315–341. Google Scholar

• [24]

I. S. Pop, F. Radu and P. Knabner, Mixed finite elemets for the Richards’ equations: Linearization procedure, J. Comput. Appl. Math. 168 (2004), 365–373.

• [25]

F. Radu, I. S. Pop and S. Attinger, Analysis of an Euler implicit-mixed finite element scheme for reactive solute transport in porous media, Numer. Methods Partial Differential Equations 26 (2010), no. 2, 320–344.

• [26]

F. Radu, I. S. Pop and P. Knabner, Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation, SIAM J. Numer. Anal. 42 (2004), 1452–1478.

• [27]

F. Radu, I. S. Pop and P. Knabner, Error estimates for a mixed finite element discretization of some degenerate parabolic equations, Numer. Math. 109 (2008), 285–311.

• [28]

K. R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solid, Math. Models Methods Appl. Sci. 17 (2007), 215–252.

• [29]

L. A. Richards, Capillary conduction of liquids through porous mediums, AIP J. Appl. Phys. 1 (1931), 318–333. Google Scholar

• [30]

Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7 (1986), no. 3, 856–869.

• [31]

M. Schatzman, Analyse numérique, Dunod, Paris, 1991. Google Scholar

• [32]

E. Schneid, P. Knabner and F. Radu, A priori error estimates for a mixed finite element discretization of the Richards’ equation, Numer. Math. 98 (2004), 353–370.

• [33]

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

• [34]

P. Sochala, A. Ern and S. Piperno, Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows, Comput. Methods Appl. Mech. Engrg. 198 (2009), 2122–2136.

• [35]

P. Sochala, A. Ern and S. Piperno, Numerical methods for subsurface flows and coupling with surface runoff, in preparation. Google Scholar

• [36]

C. S. Woodward and C. N. Dawson, Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media, SIAM J. Numer. Anal. 37 (2000), 701–724.

• [37]

D. Yakoubi, Analyse et mise en øuvre de nouveaux algorithmes en méthodes spectrales, Ph.D. thesis, Université Pierre et Marie Curie, Paris, 2007. Google Scholar

Revised: 2017-03-26

Accepted: 2017-03-29

Published Online: 2017-05-11

Published in Print: 2018-01-01

Citation Information: Advances in Pure and Applied Mathematics, Volume 9, Issue 1, Pages 1–27, ISSN (Online) 1869-6090, ISSN (Print) 1867-1152,

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.