Thesis Summary:

In this thesis, new Banach spaces-based mixed finite element methods are explored to address coupled diffusion problems and related models in continuous mechanics. The focus is on numerical analysis and simulation of the stress-assisted diffusion problem and the chemotaxis-Navier-Stokes problem. First, we introduce and analyze mixed variational formulations based on Banach spaces for the nearly incompressible linear elasticity problem and the Stokes problem. This approach is motivated by the similarities between the variational formulations of these models with respect to those obtained for the stress-assisted diffusion problem, which will be subsequently studied. To avoid the imposition of weak symmetry on the Cauchy stress tensor, we reformulate the problems in terms of the pseudostress tensor. We apply integration by parts formulas appropriate for the Banach spaces used, resulting in continuous schemes for both models. We employ the Babuška-Brezzi theory in Banach spaces and generalize classic results to establish that the obtained formulations are well-posed within these spaces. Next, we address the system of partial differential equations describing the diffusion of a solute in an elastic material. The elasticity model, whose momentum equation includes a source term dependent on diffusion, is reformulated using the non-symmetric pseudostress tensor and the deformation of the solid as unknowns of the mixed scheme. The diffusion equation, with the diffusivity function and source term depending on the stress and strain tensor of the solid, respectively, is approached using a primal formulation with concentration as the unknown. Dirichlet boundary conditions are considered for both equations. As a natural continuation of the above, a fully-mixed approach based on Banach spaces is proposed and analyzed, generating a new finite element method for the coupled stress-assisted diffusion problem to be solved numerically. We introduce two mixed schemes for the diffusion problem, using diffusion flux as an additional variable, and for the second, we also consider the concentration gradient as an unknown. Finally, we introduce and analyze a fully-mixed method based on Banach spaces to numerically solve the stationary chemotaxis-Navier-Stokes problem. This coupled and nonlinear model represents the biological process driven by cellular movements induced by an external or internal chemical signal within an incompressible fluid. In addition to the velocity and pressure of the fluid, the velocity gradient and the Bernoulli-type stress tensor are introduced as additional variables, allowing the fluid pressure to be eliminated from the equations and calculated by post-processing after solving the system. In turn, in addition to the cellular density and the concentration of the chemical signal, the pseudostresses associated with these last variables and their corresponding gradients are introduced as additional unknowns. The resulting continuous formulation, set in a Banach framework, consists of a coupled system of three saddle point problems, each perturbed with trilinear forms dependent on the data and the unknowns of the other two problems. The continuous formulations resulting from each of the schemes are approached through a fixed- point strategy. Therefore, the Babuška-Brezzi theory in Banach spaces allows us to establish that the operators associated with each of the problems are well-stated. In turn, the classic Banach fixed-point theorem, in conjunction with assumptions of small data, results in the existence and uniqueness of the solution at a continuous level. Then, on arbitrary finite element subspaces, we establish Galerkin schemes corresponding to each of the problems. Assuming that the mentioned subspaces are inf-sup stable, Brouwer’s theorem allows us to establish the existence of solutions at the discrete level. Ad- ditionally, for the scheme associated with the stationary chemotaxis-Navier-Stokes problem, Banach’s fixed-point theorem also allows establishing the uniqueness of such discrete solution. We obtain Céa’s estimates corresponding to each scheme, and once the finite element subspaces are particularized, the approximation properties allow us to establish the corresponding convergence rates. Finally, numerical experiments confirm these rates and illustrate the good performance of our methods.

ISI Publications from the Thesis

Gabriel N. GATICA, Cristian INZUNZA, Filander A. SEQUEIRA: New Banach spaces-based fully-mixed finite element methods for pseudostress-assisted diffusion problems. Applied Numerical Mathematics, vol. 193, pp. 148-178, (2023).

Sergio CAUCAO, Eligio COLMENARES, Gabriel N. GATICA, Cristian INZUNZA: A Banach spaces-based fully-mixed finite element method for the stationary chemotaxis-Navier-Stokes problem. Computers & Mathematics with Applications, vol. 145, pp. 65-89, (2023).

Gabriel N. GATICA, Cristian INZUNZA, Filander A. SEQUEIRA: A pseudostress-based mixed-primal finite element method for stress-assisted diffusion problems in Banach spaces. Journal of Scientific Computing, vol. 92, 3, article: 103, (2022).

Gabriel N. GATICA, Cristian INZUNZA: On the well-posedness of Banach spaces-based mixed formulations for the nearly incompressible Navier-Lame and Stokes equations. Computers & Mathematics with Applications, vol. 102, pp. 87-94, (2021).