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：本文针对采用增强型Taylor-Hood混合逼近离散化的不可压缩流动问题，开发了高效的预条件迭代求解器。该方法通过在常规压力空间中增加分段常数压力空间来确保局部质量守恒。当压力空间由标准Taylor-Hood基函数与分段常数压力基函数的并集定义时，这种增强过程会导致压力变量的过定，从而使得设计高效求解器变得复杂。我们首先分析了这种压力空间定义方式对相关矩阵的影响。随后，我们展示了如何通过基于奇异压力质量矩阵的预条件子恢复Stokes问题的有效求解器，以及如何采用两级压力对流扩散策略处理线性化Navier-Stokes方程导出的Oseen系统。数值实验所用代码已在线公开。