Unconditionally optimal error Estimate of a linearized Second-order Fully Discrete Finite Element Method for the bioconvection flows with concentration dependent viscosity

TL;DR

This work tackles unsteady bioconvection flows governed by a Navier–Stokes type system with concentration-dependent viscosity $\nu(c)$ coupled to a linear convection–diffusion equation for microorganism concentration $c$. It introduces second-order backward differentiation formula (BDF2) based fully discrete finite element schemes in both decoupled and fully coupled variants, and proves unconditional stability and optimal $L^2$ and $H^1$ error estimates with no time-step restrictions. The analysis employs energy methods, skew-symmetric trilinear forms to handle convection, and Stokes/Ritz projections for spatial discretization, yielding robust convergence results under suitable regularity and viscosity bounds. Numerical experiments using FreeFem++ on the unit square corroborate the theory across several viscosity models, demonstrating accurate velocity, concentration, and pressure approximations and illustrating practical applicability to bioconvection simulations.

Abstract

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 consisting of the Navier-Stokes equation coupled with 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 $L^2$-norm and $H^1$-norm are proved by using finite element approximations in space and finite differences in time. Finally, the numerical results for different viscosity are showed to support the theoretical analysis.

Figures (9)

• Figure 1: Convergence history of $(\mathbf{u},p,c)$ for $\nu=1$.
• Figure 2: Convergence history of $(\rho,\sigma,\mathbf{u},p)$ for $\nu=1+0.1c$.
• Figure 3: Convergence history of $(\rho,\sigma,\mathbf{u},p)$ for $\nu=exp(c)$.
• Figure 4: Numerical solutions of velocity at times t = 0, 0.2, 0.4, 0.6, 0.8, 1.0 with $\nu=1+0.1c$ for the decoupled finite element method.
• Figure 5: Numerical solutions of pressure at times t = 0, 0.2, 0.4, 0.6, 0.8, 1.0 with $\nu=1+0.1c$ for the decoupled finite element method.

Theorems & Definitions (8)

• Lemma 2.1
• Remark 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• Theorem 4.2