Thesis Summary:

This thesis is concerned with the numerical and mathematical analysis of new schemes to solve partial differential equations modelling fluid problems. Mainly, we focus in to develop schemes for incompressible flow and fluid in porous medium, and to compare numerical methods of pressure reconstruction strongly used in mathematical modelling of cardiovascular problems and hemodynamic. The equations involved in each chapter are analysed using classical technics of functional analysis joint finite elements methods. In adittion, we show a set of numerical examples supporting the theorical results. We start showing the mathematical and numerical analysis of the Multiscale Hybrid Mixed (MHM) Method applied to the Oseen Equation, where Lagrange multiplier is introduced as a new unknown. The goal pursued with this chapter is to lay foundations allow us in the application of MHM to the unsteady Navier-Stokes equations. Specifically, we propose and analyze a multiscale residual a posteriori error estimator. Once introduced the method is proven the solvability of the local problems by means of the Babuu ska-Brezzi theory, while a stabilized method is used like a second-level solver. What follows the above is to present the multiscale estimator two-level residual error estimator based, where we prove its efficiency and reliability. Next, we address a nonlinear Darcy equation, where the viscosity is exponentially dependent of the pressure, and using a variable change we transform a nonlinear problem in a linear problem .We introduce a new stabilized method where the parameters of stabilization of the scheme defined do not mesh dependent. Also, the scheme allows us the use of the equal polynomial order of velocity and pressure. For purposes of to analyze the convergence of the stabilized method we prove the well-posedness of the scheme and we analyze a priori analysis. In addition, an adaptive procedure is also considered, where is proven the efficiency and reliability of the proposed estimator. Finally, we devote a chapter to analyze two pressure reconstruccion methods, strongly used to estimate the pressure of the blood once the velocity field is got by means of Magnetic Resonance Imaging (MRI). There are several methods proposed in the literature but in this thesis work we consider the knowns Pressure Poisson Estimator (PPE) and Stokes Estimator (STE), which are Navier-Stokes equation (NSE) based. In both PPE and STE we consider the velocity field as a known data. While in PPE the Poisson equation is deduced applying the divergence operator in the NSE, the STE is got introducing an auxiliar velocity field free-divergence converting the NSE in a Stokes equation. We propose a estrategy for performing a priori error analysis of the reconstructed pressure fields based on the splitting of the solution in two components and we apply technicals of approximation such as continuous Galerkin and Pressure Stabilized Petrov Galerkin (PSPG). The goal is to compare the performance between these methods and to conclude about what is most cost-effective.

ISI Publications from the Thesis

Rodolfo ARAYA, Cristóbal BERTOGLIO, Cristian CáRCAMO, Sergio URIBE: Convergence analysis of pressure reconstruction methods from discrete velocities. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 57, 3, pp. 1839-1861, (2023).

Rodolfo ARAYA, Cristian CáRCAMO, Abner POZA: An adaptive stabilized finite element method for the Darcy’s equations with pressure dependent viscosities. Computer Methods in Applied Mechanics and Engineering, vol. 387, Paper no. 114100, (2021).

Rodolfo ARAYA, Cristian CáRCAMO, Abner POZA, Frederic VALENTIN: An adaptative multiscale hybrid-mixed method for the Oseen equations. Advances in Computational Mathematics, vol. 47, no. 1, Paper no. 15, (2021).