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 of the velocity and concentration in the -norm and the pressure in the -norm are derived, where and are fine and coarse mesh sizes, respectively, and denotes the iteration error with . 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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稳定生物对流的两级 Arrow-Hurwicz 迭代法


为了避免求解鞍点系统，本文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 方程和稳态平流扩散方程耦合。使用迷你单元来近似速度、压力，分段线性单元来近似浓度，我们使用线性化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线性化方案以获得细网格解。导出了 -范数中的速度和浓度以及 -范数中的压力的​​最佳误差估计，其中 和 分别是细网格和粗网格尺寸，并表示 的迭代误差。给出的数值结果支持理论分析并证实所提出的两级方法的效率。