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Mathematics of Computation

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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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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