Thesis Summary:

The aim of this thesis is to analyse, develop and implement several mathematical and numerical techniques, based on mixed finite element methods and fixed-point strategies, with the purpose of establishing the solvability of linear and non-linear problems arising in the context of fluid mechanics, more precisely, coupled problems in porous media and non-isotermal flows. We firstly derive an augmented fully-mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Navier-Stokes equations (with nonlinear viscosity) 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 the nonlinearity involved in the Navier-Stokes region is handled by setting 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. Furthermore, since the convective term in the fluid forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin redundant terms. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. We also derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. Next, we discuss the analysis of a mixed finite element method for the coupling problem of Navier-Stokes/Darcy-Forchheimer with constant density and viscosity. We consider the standard mixed formulation in the Navier-Stokes domain and the dual-mixed one in the Darcy-Forchheimer region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The well-posedness of the problem is achieved combining a fixed point strategy, classical results on nonlinear monotone operators and the well-known Schauder and Banach theorems. In particular, we employs Bernardi-Raugel and Raviart-Thomas elements for the velocities, and piecewise constant elements for the pressures and the Lagrange multiplier. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, we close this thesis with the a priori and a posteriori error analysis of an augmented fully-mixed finite element method for the non-isothermal Oldroyd-Stokes problem. For convenience of the analysis, the strain, the vorticity, and the stress tensors are introduced as further unknowns (besides the polymeric part of the extra-stress tensor, the velocity, the pressure, and the temperature of the fluid). This allows to join the polymeric and solvent viscosities in an adimensional viscosity, and to eliminate the polymeric part of the extra-stress tensor and the pressure from the system, which, together with the solvent part of the extra-stress tensor, are easily recovered later on through suitable postprocessing formulae. In this way, a fully mixed approach is applied, in which the heat flux vector is incorporated as an additional unknown as well. Furthermore, since the convective term in the heat equation forces both the velocity and the temperature to live in a smaller space than usual, we augment the variational formulation by using suitable Galerkin redundant terms. We prove solvability of both the continuous and discrete problems, with its corresponding a priori estimate. Regarding the a posteriori error analysis, two reliable and efficient residual-based estimators are derived. For all the problems described above, several numerical experiments are provided which illustrate the good performance of the proposed methods and confirm the theoretical results of convergence as well as reliability and efficiency of the respective a posteriori error estimators.

ISI Publications from the Thesis

Sergio CAUCAO, Marco DISCACCIATI, Gabriel N. GATICA, Ricardo OYARZúA: A conforming mixed finite element method for the Navier-Stokes/Darcy-Forchheimer coupled problem. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 54, 5, pp. 1689-1723, (2020).

Sergio CAUCAO, Gabriel N. GATICA, Ricardo OYARZúA: A posteriori error analysis of an augmented fully-mixed formulation for the non-isothermal Oldroyd-Stokes problem. Numerical Methods for Partial Differential Equations, vol. 35, 1, pp. 295-324, (2019).

Sergio CAUCAO, Gabriel N. GATICA, Ricardo OYARZúA: Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations. Mathematical Modelling and Numerical Analysis, vol. 52, 5, pp. 1947-1980, (2018).

Sergio CAUCAO, Gabriel N. GATICA, Ricardo OYARZúA, Ivana SEBESTOVA: A fully-mixed finite element method for the Navier-Stokes/Darcy coupled problem with nonlinear viscosity. Journal of Numerical Mathematics, vol. 25, 2, pp. 55-88, (2017).

Sergio CAUCAO, Gabriel N. GATICA, Ricardo OYARZúA: A posteriori error analysis of a fully-mixed formulation for the Navier-Stokes/Darcy coupled problem with nonlinear viscosity. Computer Methods in Applied Mechanics and Engineering, vol. 315, pp. 943-971, (2017).