Unconditionally Optimal Error Estimate of a Linearized Second-Order Fully Discrete Finite Element Method for Bioconvection Flows with Concentration-Dependent Viscosity

In this paper, the coupled and decoupled BDF2 finite element discrete schemes are obtained for the time-dependent bioconvection flows problem with concentration dependent viscosity, which consists of the Navier-Stokes equations and a linear convection-diffusion equation modeling the concentration of microorganisms in a culture fluid. The unconditionally optimal error estimate for the velocity and concentration in L²-norm are proved by using finite element approximations in space and finite differences in time. Finally, the numerical results for different viscosity are shown to support the theoretical analysis.