Two-level Arrow–Hurwicz iteration methods for the steady bio-convection flows

To avoid solving a saddle-point system, in this paper, we study two-level Arrow–Hurwicz finite element methods for the steady bio-convection flows problem which is coupled by the steady Navier–Stokes equations and the steady advection–diffusion equation. Using the mini element to approximate the velocity, pressure, and the piecewise linear element to approximate the concentration, we use the linearized Arrow–Hurwicz iteration scheme to obtain the coarse mesh solution and use three different one-step Stokes/Oseen/Newton linearized scheme to obtain the fine mesh solution. The optimal error estimate $O(h+H^2+{\chi }^{m/2})$ of the velocity and concentration in the $H^1$-norm and the pressure in the $L^2$-norm are derived, where $h$ and $H$ are fine and coarse mesh sizes, respectively, and ${\chi }^{m/2}$ denotes the iteration error with $0<\chi <1$. Numerical results are given to support the theoretical analysis and confirm the efficiency of the proposed two-level methods.