Analysis of an augmented fully-mixed finite element method for a bioconvective flows model

Abstract In this paper we study a stationary generalized bioconvection problem given by a Navier–Stokes type system coupled to a cell conservation equation for describing the hydrodynamic and micro-organisms concentration, respectively, of a culture fluid, assumed to be viscous and incompressible, and in which the viscosity depends on the concentration. The model is rewritten in terms of a first-order system based on the introduction of the shear-stress, the vorticity, and the pseudo-stress tensors in the fluid equations along with an auxiliary vector in the concentration equation. After a variational approach, the resulting weak model is then augmented using appropriate redundant parameterized terms and rewritten as fixed-point problem. Existence and uniqueness results for both the continuous and the discrete scheme as well as the respective convergence result are obtained under certain regularity assumptions combined with the Lax–Milgram theorem, and the Banach and Brouwer fixed-point theorems. Optimal a priori error estimates are derived and confirmed through some numerical examples that illustrate the performance of the proposed technique.