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

We propose and analyze a new mixed finite element method for the nonlinear problem given by coupling of steady Brinkman–Forchheimer double-diffusion equations. Besides velocity, temperature, concentration, our approach introduces velocity gradient, pseudostress tensor, pair vectors involving temperature/concentration, its gradient as further unknowns. As consequence, we obtain fully variational formulation presenting Banach spaces framework in each set In this way, differently from techniques previously developed related coupled problems, no augmentation procedure needs to be incorporated now into nor solvability analysis. The resulting non-augmented scheme is then written equivalently fixed-point equation, so that well-known theorem, combined with classical results on monotone operators Babuška–Brezzi’s theory spaces, are applied prove unique continuous discrete systems. Appropriate subspaces satisfying required inf-sup conditions specified, optimal priori error estimates derived. Several numerical examples confirm theoretical rates convergence illustrate performance flexibility method.