Preprint 2026-16
Gabriel N. Gatica, Salim Meddahi, Kevin W. Pucha-Atan, Ricardo Ruiz-Baier:
A Banach spaces-based fully mixed finite element method for the thermo-electro-hydrodynamic Boussinesq problem
Abstract:
In this work, we introduce and analyze a fully mixed finite element method for the stationary thermo--electro--hydrodynamic \rrb{(TEHD)} Boussinesq problem. The model consists of a nonlinear, strongly coupled system of equations for the velocity, pressure, temperature, and electric potential. We rewrite the governing equations as a first‑order system by introducing additional variables. This reformulation enables a consistent and stable treatment of the coupling that arises from both convective effects and electric body forces. The resulting continuous formulation consists of three nonlinear saddle‑point problems, which we decouple by introducing suitable linearizations of the corresponding variational equations. We analyze the resulting subproblems using the Babu\v{s}ka--Brezzi theory and the Lax--Milgram theorem in the Banach space setting. This analysis yields a unique admissible choice of Lebesgue exponents for the spaces of unknowns and test functions, and it also restricts the study to two dimensions. Under additional regularity for one of the subproblems and standard small‑data conditions, we then apply Banach's fixed‑point theorem to establish the unique solvability of the fully coupled problem. A discrete version of this strategy, combined with Brouwer's theorem, allows us to prove---under appropriate hypotheses on the discrete spaces---the existence of a solution to the associated Galerkin scheme. We derive the corresponding C\'ea estimate and introduce specific finite element subspaces that satisfy the required assumptions. Finally, we present numerical experiments to illustrate the performance of the method and to confirm the predicted convergence rates.
