When studying the dynamics of incompressible fluids in bounded domains the only available data often provide average flow rate conditions on portions of the domain's boundary. In engineering applications a common practice to complete these conditions is to prescribe a Dirichlet condition by assuming a-priori a spatial profile for the velocity field. However, this strongly influence the accuracy of the numerical solution. A more mathematically sound approach is to prescribe the flow rate conditions using Lagrange multipliers, resulting in an augmented weak formulation of the Navier-Stokes problem. In this paper we start from the SIMPLE preconditioner, introduced so far for the standard Navier-Stokes equations, and we derive two preconditioners for the monolithic solution of the augmented problem. This can be useful in complex applications where splitting the computation of the velocity/pressure and Lagrange multipliers numerical solutions can be very expensive. In particular, we investigate the numerical performance of the preconditioners in both idealized and real-life scenarios. Finally, we highlight the advantages of treating flow rate conditions with a Lagrange multipliers approach instead of prescribing a Dirichlet condition.

翻译：在研究有界域内不可压缩流体动力学时，唯一可用的数据通常仅提供域边界部分区域上的平均流量条件。在工程应用中，通常通过预先假设速度场的空间分布来施加狄利克雷边界条件以完善这些条件。然而，这种做法会严重影响数值解的精度。一种数学上更严谨的方法是使用拉格朗日乘子来施加流量边界条件，从而得到纳维-斯托克斯问题的增广弱形式。本文从目前针对标准纳维-斯托克斯方程提出的SIMPLE预条件子出发，推导出两种用于增广问题整体求解的预条件子。这在复杂应用中非常有用，因为在那些场景中分别计算速度/压力与拉格朗日乘子的数值解可能计算代价极高。我们特别研究了这些预条件子在理想化场景和实际应用场景中的数值性能。最后，我们阐明了采用拉格朗日乘子方法处理流量边界条件相较于直接施加狄利克雷边界条件的优势。