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+H2+χm/2) of the velocity and concentration in the H1-norm and the pressure in the L2-norm are derived, where h and H are fine and coarse mesh sizes, respectively, and χm/2 denotes the iteration error with 0<χ<1. Numerical results are given to support the theoretical analysis and confirm the efficiency of the proposed two-level methods.

中文翻译：

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用于稳态生物对流的两级 Arrow-Hurwicz 迭代方法


为避免求解鞍点系统，在本文中，我们研究了稳态生物对流问题的两级 Arrow-Hurwicz 有限元方法，该问题由稳态 Navier-Stokes 方程和稳态对流-扩散方程耦合。使用微型单元来近似速度、压力，使用分段线性单元来近似浓度，我们使用线性化的 Arrow-Hurwicz 迭代方案来获得粗网格解，并使用三种不同的一步 Stokes/Oseen/Newton 线性化方案来获得细网格解。推导了 H1 范数中速度和浓度以及 L2 范数中压力的最优误差估计 O（h+H2+χm/2），其中 h 和 H 分别为细网格尺寸和粗网格尺寸，χm/2 表示迭代误差为 0<χ<1。给出了数值结果以支持理论分析并验证了所提出的两能级方法的有效性。