Banach spaces-based analysis of a fully-mixed finite element method for the steady-state model of fluidized beds

In this paper we propose and analyze a fully-mixed finite element method for the steady-state model of fluidized beds. This numerical technique, which arises from the use of a dual-mixed approach in each phase, is motivated by a methodology previously applied to the stationary Navier–Stokes equations and related models. More precisely, we modify the stress tensors of the fluid and solid phases by defining pseudostresses as phasic stresses that include shear, pressure, and convective effects. Next, we eliminate the pressures from the equations and derive constitutive relations depending only on the aforementioned pseudostresses and the velocities of the fluid and the particles. In this way, these variables, together with the skew-symmetric parts of the velocity gradients, also named vorticities, become the only unknowns of our variational formulation. As usual, the latter is obtained by testing against suitable functions, and then integrating and integrating by parts, respectively, the equilibrium and the constitutive equations. The particle pressure, a known function of the concentration, is given as a datum, and the fluid pressure is computed afterwards via a postprocessing formula. The continuous setting, lying in a Banach spaces framework rather than in a Hilbertian one, is rewritten as an equivalent fixed-point equation, and hence the well-posedness analysis is carried out by combining the Babusˇka–Brezzi theory, the Banach–Necˇas–Babusˇka Theorem, and the classical Banach fixed-point Theorem. Thus, existence of a unique solution in a closed ball is guaranteed for sufficiently small data. In turn, the associated Galerkin scheme is introduced and analyzed analogously, so that, under suitable assumptions on generic finite element subspaces, and for sufficiently small data as well, the Brouwer and Banach fixed-point Theorems allow to conclude existence and uniqueness of solution, respectively. Specific finite element subspaces satisfying the required hypotheses are described, and optimal a priori error estimates are derived. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence, are reported.