Métodos de elementos finitos mixtos para problemas no lineales en biomedicina y biología = Mixed finite element methods for nonlinear problems in biomedicine and biology.

Abstract

This thesis aims to develop the mathematical and numerical analysis of nonlinear coupled partial differential equations (PDE’s)-based models that describe certain phenomena in Biology and Biomedicine encompassing generalized bioconvection and deformable image registration. More precisely, we introduce primal and mixed schemes based on finite elements for the aforementioned models, prove the solvability of the continuous and discrete problems, establish the corresponding error estimates, and present a variety of tests to validate the theoretical results and illustrate the performance of such methods including applied examples. We begin with the bioconvective flows model, which describes the hydrodynamics of microorganisms in a culture fluid and takes place in several biological processes, including reproduction, infection, and the marine life ecosystem. The flows are governed by a Navier-Stokes type system coupled to a conservation equation that models the microorganisms concentration. The culture fluid is assumed to be viscous and incompressible with a concentration dependent viscosity. For the mathematical analysis, the model is rewritten in terms of a first-order system based on the introduction of the strain, the vorticity, and the pseudo-stress tensors in the fluid equations along with an auxiliary vector in the concentration equation. The resulting weak model is then augmented using appropriate redundant parameterized terms and rewritten as a fixed-point problem. Existence and uniqueness results for both the continuous and the discrete scheme are obtained under certain regularity assumptions combined with the Lax-Milgram theorem or the Babuˇska-Brezzi theory, and the Banach and Brouwer fixed-point theorems. Optimal a priori error estimates are also derived and confirmed via numerical examples. Next, we address the study of a deformable image registration (DIR) model, which arises in numerous research fields as a solution to the combination or comparison of a series of images. Specifically, in Biomedicine, there is a need to detect changes in images obtained from the same subject over time, whereby the deformable image registration represents a powerful computational method for image analysis, with promising applications in the diagnosis of human disease. One important and recent application of DIR is the study of local lung tissue deformation from computed-tomography images of the thorax, which allows the early detection of damage induced by mechanical ventilation in the lung. In our case, for the first model studied in this part, which we will call extended deformable image registration problem, we propose a finite element method for its numerical approximation, proving well-posedness of the primal and dual-mixed continuous formulations, as well as of the associated Galerkin schemes. A priori error estimates and the corresponding rates of convergence are also established for both discrete methods. In addition, we provide numerical examples confronting our formulations with the standard ones. Finally, in order to guarantee an appropriate convergence behavior of the discrete approximations