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# Graduate Thesis of Ricardo Oyarzúa

ProgramPhD in Applied Sciences with mention in Mathematical Engineering, Universidad de Concepción
Enrollment Year2007
Senior Year2010
Thesis TitleMixed Finite Element Methods for the Stokes-Darcy Coupling Problem

#### Thesis Summary:

The main purpose of this thesis is to approximate a coupling of fluid flow with porous medium flow by using Mixed Finite Element Methods. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding interface conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. First, we analyze the well-posedness of a mixed formulation, primal in the Stokes domain and dual-mixed in the Darcy region, and we show that use of any pair of stable Stokes and Darcy elements implies the well-posedness of the corresponding Stokes-Darcy Galerkin scheme. This extends previous results showing well-posedness only for Bernardi-Raugel and Raviart-Thomas elements of the lowest order. Afterwards, we develop the a priori and a posteriori error analysis of a new fully mixed finite element method for the coupled problem. We consider dual-mixed formulations in both domains, which yields the pseudostress and the velocity in the fluid, as well as the velocity and the pressure in the porous medium, as the main unknowns. In addition, since the transmission conditions become essential, we impose them weakly and introduce the values of the porous medium pressure and the fluid velocity on the interface as new unknowns, that play the role of Lagrange multipliers. We prove the unique solvability of the continuous formulation and derive sufficient conditions on the finite element subspaces ensuring that the associated Galerkin scheme is well posed. A practicable choice of subspaces is given by the Raviart-Thomas elements of lowest order and piecewise constants for the velocities and pressures, respectively, in both domains, together with continuous piecewise linear elements for the Lagrange multipliers. We also derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. Finally, we generalize the above results and we analyze a mixed finite element method for the coupling of viscous incompressible fluid flow with a state law mathematically corresponding to a nonlinear porous medium flow. Flows are governed by the Stokes and nonlinear Darcy equations respetively. In the latter permeability is given by a strongly monotone and Lipschitz-continuous nonlinear operator. We consider dual-mixed approaches in both the Stokes and Darcy regions. This yields a twofold saddle point operator equation as the resulting variational formulation. A well known generalization of the classical Babuˇska-Brezzi theory is applied to show the wellposedness of the continuous and discrete formulations and to derive the corresponding a priori error estimate. Furthermore, a reliable and efficient residual based a posteriori error estimator is provided. For all the situations described above, several numerical results illustrating the correct performance of the method, and confirming the theoretical results, are reported.

Thesis Director(s) Gabriel N. Gatica, Francisco-Javier Sayas
Thesis Project Approval Date2008, July 17
Thesis Defense Date2010, December 15
Professional MonitoringAs of March 2011, Assistant Professor of the Departamento de Matemática of the Universidad del Bío Bío, Concepción.

#### ISI Publications from the Thesis

Gabriel N. GATICA, Ricardo OYARZúA, Francisco J. SAYAS: A twofold saddle point approach for the coupling of fluid flow with nonlinear porous media flow. IMA Journal of Numerical Analysis, vol. 32, 3, pp. 845-887, (2012).

Gabriel N. GATICA, Ricardo OYARZúA, Francisco J. SAYAS: Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem. Mathematics of Computation, vol. 80, 276, pp. 1911-1948, (2011).

Gabriel N. GATICA, Ricardo OYARZúA, Francisco J. SAYAS: Convergence of a family of Galerkin discretizations for the Stokes-Darcy coupled problem. Numerical Methods for Partial Differential Equations, vol. 27, 3, pp. 721-748, (2011).

Gabriel N. GATICA, Ricardo OYARZúA, Francisco J. SAYAS: A residual-based a posteriori error estimator for a fully-mixed formulation of the Stokes-Darcy coupled problem. Computer Methods in Applied Mechanics and Engineering, vol. 200, 21-22, pp. 1877-1891, (2011).

#### Other Publications (ISI)

Gabriel N. GATICA, Salim MEDDAHI, Ricardo OYARZúA: A conforming mixed finite-element method for the coupling of fluid flow with porous media flow. IMA Journal of Numerical Analysis, vol. 29, 1, pp. 86-108, (2009)

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