Eligio Colmenares, Gabriel N. Gatica, Sebastian Moraga:
A Banach spaces-based analysis of a new fully-mixed finite element method for the Boussinesq problem
In this paper we propose and analyze, utilizing mainly tools and abstract results from Banach spaces rather than from Hilbert ones, a new fully-mixed finite element method for the stationary Boussinesq problem with temperature-dependent viscosity. More precisely, following an idea that has already been applied to the Navier-Stokes equation and to the fluid part only of our model of interest, we first incorporate the velocity gradient and the associated Bernoulli stress tensor as auxiliary unknowns of the fluid. Additionally, and differently from earlier works in which either the primal or the classical dual-mixed method is employed for the heat equation, we consider here an analogue of the approach for the fluid, which consists of introducing as further variables the gradient of temperature and a vector version of the Bernouilli tensor. The resulting mixed variational formulation, which involves the aforementioned four unknowns together with the original variables given by the velocity and temperature of the fluid, is then reformulated as a fixed point equation. Next, we utilize the well-known Banach and Brouwer theorems, combined with the application of the Babuska-Brezzi theory to each independent equation, to analyze the solvability of the continuous and discrete schemes. In particular, Raviart-Thomas spaces of order k \ge n-1 for the Bernoulli tensor and its vector version for the heat equation, and piecewise polynomials of degree \le k for the velocity, the temperature, and both gradients, become a feasible choice. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the performance of the fully-mixed scheme and confirming the theoretical rates of convergence.