Eligio Colmenares, Gabriel N. Gatica, Willian Miranda:
Analysis of an augmented fully-mixed finite element method for a bioconvective flows model
In this paper we study a stationary generalized bioconvection problem given by the Navier-Stokes equations coupled to a cell conservation equation for describing the hydrodynamic and micro-organisms concentration of a culture fluid, assumed to be viscous and incompressible, and in which the viscosity might depend 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 parameterized Galerkin terms and rewritten as fixed-point problem. Existence, uniqueness and convergence results 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.