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Undergraduate Thesis of Hugo Díaz

Díaz, HugoCareerMathematical Civil Engineering, Universidad de Concepción
Enrollment Year2006
Senior Year2015
Thesis TitleThe Darcy-Forchheimer and Stokes Coupling Problem

Thesis Summary:

The purpose of this thesis is to extend previous results concerning the analysis of transmission problems in fluid mechanics to the case of a fully-mixed formulation for the nonlinear model given by the Darcy - Forchheimer / Stokes coupled problem with the Saffman condition at the interface. Differently from the usual tools available in the literature, which are valid mainly for Hilbertian structures, our aproach is applicable to Banach spaces, which is precisely the case of the present nonlinear model. In addition, the appropriate choice of the discrete spaces allows, for example, the use of lowest order Raviart-Thomas interpolator directly to the velocity, thus avoiding the use of liftings to analyse the inf-sup conditions, and hence the hypothesis of quasiuniformity in a neighborhood of the interface on the porous medium side is not needed anymore. This work is organized as follows. In Chapter 2 we introduce the notation and the main aspects of the continuous Darcy - Forchheimer / Stokes coupled problem, which includes the weak formulation and the identification of the resulting system structure as a twofold saddle point problem. The analysis of a modified abstract theory for this kind of nonlinear operator equations, including the continuous and discrete setting, is analyzed in Chapter 3. The results of this chapter are then applied to our model problem in Chapter 4, and specific finite element subspaces satisfying the required conditions are defined. Next, in Chapter 5 we consider a more general problem which consists of a nonlinear version of the Stokes model in the free fluid part, and finally, some auxiliary results are detailed in the Appendices.

Thesis Director(s) Gabriel N. Gatica
Thesis Project Approval Date2013, April 01
Thesis Defense Date2015, September 02
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ISI Publications from the Thesis

Javier A. ALMONACID, Hugo S. DíAZ, Gabriel N. GATICA, Antonio MARQUEZ: A fully-mixed finite element method for the coupling of the Stokes and Darcy-Forchheimer problems. IMA Journal of Numerical Analysis, vol. 40, 2, pp. 1454-1502, (2020).

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