A new mixed-FEM for steady-state natural convection models allowing conservation of momentum and thermal energy

A fully-mixed formulation in Banach spaces for the coupling of the steady Brinkman–Forchheimer and double-diffusion equations

“…We point out that, differently from [13, Section 6, Example 3], Dirichlet boundary conditions for temperature and concentration are assumed on the top and bottom of the domain instead of on the left and right sides of Ω as in [13]. We also note that, using similar arguments to those employed in [14], we are able to extended our analysis to the present case of mixed boundary conditions for the double-diffusion equations. In Figure 6.3, we display the computed magnitude of the velocity, velocity gradient, pseudostress tensor, and gradients of the temperature and concentration, and the temperature and concentration fields, which were built using the fully-mixed P 0 − P 0 − RT 0 − P 0 − P 0 − RT 0 approximation on a mesh with 27, 287 triangle elements (actually representing 475, 313 DOF).…”

“…In order to overcome this, in recent years there has arisen an increasing development on Banach spaces-based mixed finite element methods to solve a wide family of single and coupled nonlinear problems in continuum mechanics (see, e.g. [5], [6], [10], [11], [14], [15], [17], and [18]). This kind of procedures shows two advantages at least: no augmentation is required, and the spaces to which the unknowns belong are the natural ones arising from the application of the Cauchy-Schwarz and Hölder inequalities to the terms resulting from the testing and integration by parts of the equations of the model.…”