Pre-Publicación 2025-14
Gabriel N. Gatica, Zeinab Gharibi, Ricardo Oyarzúa:
Banach spaces-based fully mixed finite element methods for the n-dimensional Boussinesq problem with temperature-dependent parameters
Abstract:
In this paper, we introduce and analyze a family of mixed finite element methods for the numerical solution of heat-driven flows with temperature-dependent parameters, modeled by a generalization of the stationary Boussinesq equations. Our approach relies on a reformulation of the governing equations in terms of the velocity, strain-rate tensor, vorticity, stress, pseudoheat, temperature, and its gradient. The pressure is eliminated from the system using the incompressibility constraint and can be subsequently recovered through a postprocessing formula involving the stress and velocity fields. Then, the resulting continuous formulation consists of a Banach spaces-based nonlinearly perturbed coupled system of twofold saddle point operator equations. By introducing suitable linearizations of the corresponding variational equations, we establish the unique solvability of the continuous problem through a fixed-point strategy. This analysis combines the Banach--Ne\v{c}as--Babu\v{s}ka and Babu\v{s}ka--Brezzi theories in Banach spaces with the Banach fixed-point theorem, under an extraregularity assumption on the aforementioned linear systems and a smallness assumption on the data. Adopting an analogous approach for the associated Galerkin scheme, and under suitable hypotheses on the finite element subspaces employed, we establish existence of a discrete solution by applying the Brouwer fixed-point theorem and the discrete versions of the Banach--Ne\v{c}as--Babu\v{s}ka and Babu\v{s}ka--Brezzi theories. Furthermore, the error analysis is carried out under appropriate assumptions on the data, and by employing similar arguments to those yielding Strang-type estimates. Finally, several numerical experiments are presented to illustrate the performance of the proposed scheme and to confirm the convergence rates predicted by the theoretical analysis.
