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A mixed finite element method for Darcy's equations with pressure dependent porosity

Authors: Gabriel N. Gatica, Ricardo Ruiz-Baier and Giordano Tierra
Journal: Math. Comp. 85 (2016), 1-33
MSC (2010): Primary 65N15, 65N30, 74F10, 74S05
Published electronically: June 8, 2015
MathSciNet review: 3404441
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Abstract: In this work we develop the a priori and a posteriori error analyses of a mixed finite element method for Darcy's equations with porosity depending exponentially on the pressure. A simple change of variable for this unknown allows us to transform the original nonlinear problem into a linear one whose dual-mixed variational formulation falls into the frameworks of the generalized linear saddle point problems and the fixed point equations satisfied by an affine mapping. According to the latter, we are able to show the well-posedness of both the continuous and discrete schemes, as well as the associated Cea estimate, by simply applying a suitable combination of the classical Babuška-Brezzi theory and the Banach fixed point theorem. In particular, given any integer $ k \ge 0$, the stability of the Galerkin scheme is guaranteed by employing Raviart-Thomas elements of order $ k$ for the velocity, piecewise polynomials of degree $ k$ for the pressure, and continuous piecewise polynomials of degree $ k + 1$ for an additional Lagrange multiplier given by the trace of the pressure on the Neumann boundary. Note that the two ways of writing the continuous formulation suggest accordingly two different methods for solving the discrete schemes. Next, we derive a reliable and efficient residual-based a posteriori error estimator for this problem. The global inf-sup condition satisfied by the continuous formulation, Helmholtz decompositions, and the local approximation properties of the Raviart-Thomas and Clément interpolation operators are the main tools for proving the reliability. In turn, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of both methods, confirming the aforementioned properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are reported.

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  • [1] Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, NJ, Toronto, London, 1965. MR 0178246 (31 #2504)
  • [2] Etienne Ahusborde, Mejdi Azaïez, Faker Ben Belgacem, and Christine Bernardi, Automatic simplification of Darcy's equations with pressure dependent permeability, ESAIM Math. Model. Numer. Anal. 47 (2013), no. 6, 1797-1820. MR 3123377,
  • [3] Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Engrg. 142 (1997), no. 1-2, 1-88. MR 1442375 (98e:65092),
  • [4] A. Alonso, Error estimators for a mixed method, Numer. Math. 74 (1996), no. 4, 385-395. MR 1414415 (97g:65212),
  • [5] Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742-760. MR 664882 (83f:65173),
  • [6] Mejdi Azaïez, Faker Ben Belgacem, Christine Bernardi, and Nejmeddine Chorfi, Spectral discretization of Darcy's equations with pressure dependent porosity, Appl. Math. Comput. 217 (2010), no. 5, 1838-1856 (English, with English and French summaries). MR 2727929 (2011i:76061),
  • [7] 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 methods for Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 166 (1998), no. 3-4, 183-199. MR 1659191 (99j:76070),
  • [8] Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1-359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106 (54 #9111)
  • [9] Ivo Babuška and Gabriel N. Gatica, On the mixed finite element method with Lagrange multipliers, Numer. Methods Partial Differential Equations 19 (2003), no. 2, 192-210. MR 1958060 (2004b:65174),
  • [10] Tomás P. Barrios, Gabriel N. Gatica, María González, and Norbert Heuer, A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity, M2AN Math. Model. Numer. Anal. 40 (2006), no. 5, 843-869 (2007). MR 2293249 (2008b:74039),
  • [11] Barus, C., Isotherms, isopiestics and isometrics relative to viscosity. American Journal of Science, 45 (1893) 87-96.
  • [12] Christine Bernardi, Claudio Canuto, and Yvon Maday, Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem, SIAM J. Numer. Anal. 25 (1988), no. 6, 1237-1271. MR 972452 (90e:65151),
  • [13] Christine Bernardi, Frédéric Hecht, and Olivier Pironneau, Coupling Darcy and Stokes equations for porous media with cracks, M2AN Math. Model. Numer. Anal. 39 (2005), no. 1, 7-35. MR 2136198 (2006a:76107),
  • [14] D. Braess and R. Verfürth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal. 33 (1996), no. 6, 2431-2444. MR 1427472 (97m:65201),
  • [15] Franco Brezzi and Michel Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • [16] Carsten Carstensen, An a posteriori error estimate for a first-kind integral equation, Math. Comp. 66 (1997), no. 217, 139-155. MR 1372001 (97e:65147),
  • [17] Carsten Carstensen, A posteriori error estimate for the mixed finite element method, Math. Comp. 66 (1997), no. 218, 465-476. MR 1408371 (98a:65162),
  • [18] J. Chang, and K. B. Nakshatrala, Modification to Darcy model for high pressure and high velocity applications and associated mixed finite element formulations. arXiv:1306.5216[cs.NA]
  • [19] Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174 (58 #25001)
  • [20] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér., RAIRO Analyse Numérique 9 (1975), no. R-2, 77-84 (English, with Loose French summary). MR 0400739 (53 #4569)
  • [21] M. Daadaa, Discrétisation Spectrale et par Éléments Spectraux des Équations de Darcy. Ph.D. Thesis, Université Pierre et Marie Curie, Paris (2009).
  • [22] H. Darcy, Les fontaines publiques de la ville de Dijon. Dalmont, Paris (1856).
  • [23] Marco Discacciati, Edie Miglio, and Alfio Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows, Appl. Numer. Math. 43 (2002), no. 1-2, 57-74. MR 1936102 (2003h:76087),
  • [24] C. Domínguez, G. N. Gatica, and S. Meddahi, A posteriori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem. Preprint 2014-02, Centro de Investigación en Ingeniería Matemática (CI$ ^2$MA), Universidad de Concepción, Concepción, Chile, (2014).
  • [25] V. J. Ervin, E. W. Jenkins, and S. Sun, Coupling nonlinear Stokes and Darcy flow using mortar finite elements, Appl. Numer. Math. 61 (2011), no. 11, 1198-1222. MR 2842139 (2012j:65398),
  • [26] Juan Galvis and Marcus Sarkis, Non-matching mortar discretization analysis for the coupling Stokes-Darcy equations, Electron. Trans. Numer. Anal. 26 (2007), 350-384. MR 2391227 (2009a:76120)
  • [27] Gabriel N. Gatica, A Simple Introduction to the Mixed Finite Element Method: Theory and Applications, Springer Briefs in Mathematics, Springer, Cham, 2014. MR 3157367
  • [28] Gabriel N. Gatica, Luis F. Gatica, and Antonio Márquez, Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow, Numer. Math. 126 (2014), no. 4, 635-677. MR 3175180,
  • [29] Gabriel N. Gatica, George C. Hsiao, and Salim Meddahi, A residual-based a posteriori error estimator for a two-dimensional fluid-solid interaction problem, Numer. Math. 114 (2009), no. 1, 63-106. MR 2557870 (2011a:65401),
  • [30] Gabriel N. Gatica and Matthias Maischak, A posteriori error estimates for the mixed finite element method with Lagrange multipliers, Numer. Methods Partial Differential Equations 21 (2005), no. 3, 421-450. MR 2128589 (2005m:65266),
  • [31] Gabriel N. Gatica, Salim Meddahi, and Ricardo Oyarzúa, A conforming mixed finite-element method for the coupling of fluid flow with porous media flow, IMA J. Numer. Anal. 29 (2009), no. 1, 86-108. MR 2470941 (2010b:76118),
  • [32] Gabriel N. Gatica, Ricardo Oyarzúa, and Francisco-Javier Sayas, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem, Math. Comp. 80 (2011), no. 276, 1911-1948. MR 2813344 (2012i:65259),
  • [33] Vivette Girault, François Murat, and Abner Salgado, Finite element discretization of Darcy's equations with pressure dependent porosity, M2AN Math. Model. Numer. Anal. 44 (2010), no. 6, 1155-1191. MR 2769053 (2012e:76126),
  • [34] Vivette Girault and Pierre-Arnaud Raviart, Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383 (88b:65129)
  • [35] V. Girault and M. F. Wheeler, Numerical discretization of a Darcy-Forchheimer model, Numer. Math. 110 (2008), no. 2, 161-198. MR 2425154 (2009g:65163),
  • [36] Ohannes A. Karakashian, On a Galerkin-Lagrange multiplier method for the stationary Navier-Stokes equations, SIAM J. Numer. Anal. 19 (1982), no. 5, 909-923. MR 672567 (83j:65107),
  • [37] Trygve Karper, Kent-Andre Mardal, and Ragnar Winther, Unified finite element discretizations of coupled Darcy-Stokes flow, Numer. Methods Partial Differential Equations 25 (2009), no. 2, 311-326. MR 2483769 (2010a:65240),
  • [38] J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications I. Dunod, Paris, 1968.
  • [39] A. Márquez, S. Meddahi, and F.-J. Sayas, Strong coupling of finite element methods for the Stokes-Darcy problem. IMA Journal of Numerical Analysis (2014) doi: 10.1093/imanum/dru023.
  • [40] William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. MR 1742312 (2001a:35051)
  • [41] K. B. Nakshatrala and K. R. Rajagopal, A numerical study of fluids with pressure-dependent viscosity flowing through a rigid porous medium, Internat. J. Numer. Methods Fluids 67 (2011), no. 3, 342-368. MR 2835720 (2012g:76171),
  • [42] K. B. Nakshatrala and D. Z. Turner, A mixed formulation for a modification to Darcy equation based on Picard linearization and numerical solutions to large-scale realistic problems, Int. J. Comput. Methods Eng. Sci. Mech. 14 (2013), no. 6, 524-541. MR 3172099,
  • [43] J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques. Mason, Paris, 1967.
  • [44] Hao Pan and Hongxing Rui, Mixed element method for two-dimensional Darcy-Forchheimer model, J. Sci. Comput. 52 (2012), no. 3, 563-587. MR 2948707,
  • [45] Eun-Jae Park, Mixed finite element methods for generalized Forchheimer flow in porous media, Numer. Methods Partial Differential Equations 21 (2005), no. 2, 213-228. MR 2114948 (2005i:76075),
  • [46] Siegfried Prössdorf and Bernd Silbermann, Numerical Analysis for Integral and Related Operator Equations, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], vol. 84, Akademie-Verlag, Berlin, 1991. MR 1206476 (94f:65126a)
  • [47] K. R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids, Math. Models Methods Appl. Sci. 17 (2007), no. 2, 215-252. MR 2292356 (2007k:76156),
  • [48] Béatrice Rivière and Ivan Yotov, Locally conservative coupling of Stokes and Darcy flows, SIAM J. Numer. Anal. 42 (2005), no. 5, 1959-1977. MR 2139232 (2006a:76035),
  • [49] J. E. Roberts and J.-M. Thomas, Mixed and hybrid methods, Handbook of Numerical Analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 523-639. MR 1115239
  • [50] S. Srinivasan, A. Bonito, and K. R. Rajagopal, Flow of a fluid through a porous solid due to high pressure gradients. Journal of Porous Media, 16(3) (2013) 193-203.
  • [51] S. Srinivasan and K. R. Rajagopal, A thermodynamic basis for the derivation of the Darcy, Forchheimer and Brinkman models for flows through porous media and their generalizations, International Journal of Nonlinear Mechanics, 58 (2014) 162-166.
  • [52] R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), no. 3, 309-325. MR 993474 (90d:65187),
  • [53] R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math. 50 (1994), no. 1-3, 67-83. MR 1284252 (95c:65171),
  • [54] R. Verfürth, A Review of A-Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, John Wiley and Teubner Series, Advances in Numerical Mathematics 1996.
  • [55] Xiaoping Xie, Jinchao Xu, and Guangri Xue, Uniformly-stable finite element methods for Darcy-Stokes-Brinkman models, J. Comput. Math. 26 (2008), no. 3, 437-455. MR 2421892 (2009g:76087)

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Additional Information

Gabriel N. Gatica
Affiliation: CI$^{2}$MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Ricardo Ruiz-Baier
Affiliation: Institute of Earth Sciences, Quartier UNIL-Mouline, Bâtiment Géopolis, University of Lausanne, CH-1015 Lausanne, Switzerland

Giordano Tierra
Affiliation: Mathematical Institute, Faculty of Mathematics and Physics, Charles University in Prague, Prague 8, 186 75, Czech Republic

Received by editor(s): February 18, 2014
Received by editor(s) in revised form: June 11, 2014, and July 21, 2014
Published electronically: June 8, 2015
Additional Notes: The work of the first author was partially supported by CONICYT-Chile through BASAL project CMM, Universidad de Chile, and project Anillo ACT1118 (ANANUM); by the ministry of Education through the project REDOC.CTA of the Graduate School, Universidad de Concepción; and by Centro de Investigación en Ingeniería Matemática (CI$^{2}$MA), Universidad de Concepción
The work of the second author was partially supported by the University of Lausanne and by the Swiss National Science Foundation through grant PPOOP2-144922
The work of the third author was partially supported by the Ministry of Education, Youth and Sports of the Czech Republic through the ERC-CZ project LL1202. Part of this research was developed while this author was visiting CI$^{2}$MA during the last three weeks of January 2014
Article copyright: © Copyright 2015 American Mathematical Society

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