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, 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 on the matrices involved. Next, we show how to recover effective solvers for Stokes problems, with a preconditioner 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：本文针对采用富集泰勒-胡德混合近似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。该近似通过在常规压力空间中附加分片常数压力，以确保局部质量守恒。这种富集过程导致压力过指定，使得相应线性系统高效求解器的设计与实现复杂化。我们首先描述了这种压力空间选择对所涉及矩阵的影响。接着，展示了如何基于奇异压力质量矩阵恢复斯托克斯问题的有效求解器，并针对由线性化纳维-斯托克斯方程导出的奥辛系统，采用两级压力对流-扩散策略实现求解。生成数值结果的代码已在线上公开。