Sergio Caucao, Ricardo Oyarzúa, Segundo Villa-Fuentes:
A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy
In this work we present a new mixed finite element method for a class of natural convection models describing the behavior of non-isothermal incompressible fluids subject to a heat source. More precisely, we consider a system based on the coupling of the steady-state equations of momentum (Navier-Stokes) and thermal energy by means of the Boussinesq approximation. Our approach is based on the introduction of a modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the Navier–Stokes equations for the fluid and a vector unknown involving the temperature, its gradient and the velocity. The introduction of these further unknowns lead to a mixed formulation where the aforementioned pseudostress tensor and vector unknown, together with the velocity and the temperature, are the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the pseudostress tensor and the vector unknown, and discontinuous piece-wise polynomial elements of degree k for the velocity and temperature. With this choice of spaces, both, momentum and thermal energy, are conserved if the external forces belong to the velocity and temperature discrete spaces, respectively, which constitutes one of the main feature of our approach. We employ the Banach–Nečas–Babuška and Banach’s fixed point theorems to prove unique solvability for both, the continuous and discrete problems. We provide the convergence analysis and particularly prove that the error decay with optimal rate of convergence. Further variables of interest, such as the fluid pressure, the fluid vorticity, the fluid velocity gradient, and the heat-flux can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. Finally, several numerical results illustrating the performance of the method are provided.