Numerical analysis of forward and inverse fluid mechanics problems

In this thesis, we have proposed and analysed PDE-based fluid models such as Navier- Stokes, Oseen and nonlinear Darcy. We have proved their solvability using the classical Ladyzhenskaya–Babuska–Brezzi, proposed and used stabilized schemes and obtained approximated solutions by using adaptive strategies, with a strong emphasis on stabilized finite element methods. In Chapter 2 we proposed an extension of the MHM method to the Oseen equations based on previous works for the Stokes model and the advection-diffusion equation. In Chapter 3, we have introduced a new stabilized formulation for a Darcy equation with exponentially pressure-dependent porosity in two or three dimensions. We included well–posed results as a priori error estimated under standard assumptions. This new formulation allows us to use equal-order of interpolation spaces for both velocity and pressure. Besides, we introduced and studied an a posteriori error estimator of the residual type. In particular, we prove the equivalence between our error estimator and the approximation error. Chapter 4 was focused on the error analysis of methods for approximation of the pressure from discrete velocity fields, STE and PPE. The STE is analyzed using the classical Taylor-Hood finite element spaces and pressure-stabilizing Petrov-Galerkin (PSPG). The PPE is analyzed using a continuous Galerkin method, where two versions of the PPE have been considered, the classical approach without a viscous term and a new approach with a boundary viscous term.