This paper develops efficient preconditioned iterative solvers for incompressible flow problems discretised by an enriched Taylor-Hood mixed approximation, in which the usual pressure space is augmented by a piecewise constant pressure to ensure local mass conservation. This enrichment process causes over-specification of the pressure when the pressure space is defined by the union of standard Taylor-Hood basis functions and piecewise constant pressure basis functions, which complicates the design and implementation of efficient solvers for the resulting linear systems. We first describe the impact of this choice of pressure space specification on the matrices involved. Next, we show how to recover effective solvers for Stokes problems, with preconditioners based on the singular pressure mass matrix, and for Oseen systems arising from linearised Navier-Stokes equations, by using a two-stage pressure convection-diffusion strategy. The codes used to generate the numerical results are available online.

翻译：本文针对采用富集Taylor-Hood混合逼近离散的不可压缩流动问题，开发了高效的预处理迭代求解器。该方法通过在标准压力空间中添加分片常数压力基函数来确保局部质量守恒。当压力空间由标准Taylor-Hood基函数与分片常数压力基函数共同定义时，这种富集过程会导致压力超定问题，从而增加了相关线性系统求解器的设计与实现难度。我们首先描述了这种压力空间定义方式对相关矩阵的影响。接着，基于奇异压力质量矩阵的预处理方法，展示了如何恢复Stokes问题的有效求解器；同时针对纳维-斯托克斯方程线性化产生的Oseen系统，采用两阶段压力对流-扩散策略实现了高效求解。文中数值实验的代码已开源发布。