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Undergraduate Thesis of Ramiro J. Rebolledo

Rebolledo, Ramiro J.CareerMathematical Civil Engineering, Universidad de Concepción
Enrollment Year2003
Senior Year2013
Thesis TitleAnalysis of a Fully Mixed Enhanced Approach for Coupling of Quasi-Newtonian Fluids with Porous Media

Thesis Summary:

In this thesis we introduce and analyze an augmented mixed finite element method for the coupling of quasi-Newtonian fluids and porous media. The flows are governed by a class of nonlinear Stokes and linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We apply dual-mixed formulations in both domains, and, in order to handle the nonlinearity involved in the Stokes region, we set the strain and vorticity tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly, which yields the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. Moreover, in order to facilitate the analysis, we augment the formulation in the fluid by incorporating a redundant Galerkin-type term arising from the quasi-Newtonian constitutive law multiplied by a suitable stabilization parameter. In this way, under a suitable and explicit choice of this parameter, a generalization of the Babuska-Brezzi theory is utilized to show the well-posedness of the continuous and discrete formulations and to derive the corresponding a-priori error estimate. In particular, the feasible finite element subspaces include PEERS and Arnold-Falk-Winther elements for the stress, velocity and vorticity in the fluid, Raviart-Thomas elements and piecewise constants for the velocity and pressure in the porous medium, together with piecewise constant Stokes strain tensor and continuous piecewise linear elements for the traces. Next, we employ classical approaches, which include linearization techniques, Cléments interpolator and Helmholtzs decomposition, together with known efficiency estimates, to derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. Finally, several numerical results confirming the good performance of the method and the properties of the a posteriori error estimator, and illustrating the capability of the corresponding adaptive algorithm to identify the singular regions of the solution, are reported.

Thesis Director(s) Gabriel N. Gatica, Ricardo E. Oyarzua
Thesis Project Approval Date2011, May 11
Thesis Defense Date2013, July 29
Professional MonitoringRamiro Rebolledo entered the second semester of 2013 to our program of PhD in Applied Sciences with mention in Mathematical Engineering, Universidad de Concepción.
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ISI Publications from the Thesis

Gabriel N. GATICA, Antonio MARQUEZ, Ricardo OYARZúA, Ramiro REBOLLEDO: Analysis of an augmented fully-mixed approach for the coupling of quasi-Newtonian fluids and porous media. Computer Methods in Applied Mechanics and Engineering, vol. 270, 1, pp. 76-112, (2014).

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