  This work is devoted to the theoretical and numerical analysis of a
two-species chemotaxis- Navier-Stokes system with Lotka-Volterra competitive
kinetics in a bounded domain of Rd, d = 2, 3. First, we study the existence of
global weak solutions and establish a regularity criterion which provides
sufficient conditions to ensure the strong regularity of the weak solutions.
After, we propose a finite element numerical scheme in which we use a splitting
technique obtained by introducing an auxiliary variable given by the gradient
of the chemical concentration and applying an inductive strategy, in order to
deal with the chemoattraction terms in the two-species equations and prove
optimal error estimates. For this scheme, we study the well-posedness and
derive some uniform estimates for the discrete variables required in the
convergence analysis. Finally, we present some numerical simulations oriented
to verify the good behavior of our scheme, as well as to check numerically the
optimal error estimates proved in our theoretical analysis.
