For incompressible flow models, the pressure term serves as a Lagrange multiplier to ensure that the incompressibility constraint is satisfied. In engineering applications, the pressure term is necessary for calculating important quantities based on stresses like the lift and drag. For reduced order models generated via a Proper orthogonal decomposition, it is common for the pressure to drop out of the equations and produce a velocity-only reduced order model. To recover the pressure, many techniques have been numerically studied in the literature; however, these techniques have undergone little rigorous analysis. In this work, we examine two of the most popular approaches: pressure recovery through the Pressure Poisson equation and recovery via the momentum equation through the use of a supremizer stabilized velocity basis. We examine the challenges that each approach faces and prove stability and convergence results for the supremizer stabilized approach. We also investigate numerically the stability and convergence of the supremizer based approach, in addition to its performance against the Pressure Poisson method.

中文翻译：

---

基于POD的瞬态Navier--Stokes方程降阶模型的Supremizer压力恢复误差分析

对于不可压缩流动模型，压力项用作拉格朗日乘子以确保满足不可压缩约束。在工程应用中，压力项对于根据升力和阻力等应力计算重要量是必要的。对于通过适当的正交分解生成的降阶模型，压力通常会从方程中退出并生成仅速度的降阶模型。为了恢复压力，文献中已经对许多技术进行了数值研究。然而，这些技术几乎没有经过严格的分析。在这项工作中，我们研究了两种最流行的方法：通过压力泊松方程恢复压力和通过使用超稳定速度基础通过动量方程恢复。我们检查了每种方法面临的挑战，并证明了 supremizer 稳定方法的稳定性和收敛结果。除了针对压力泊松方法的性能外，我们还从数值上研究了基于 supremizer 的方法的稳定性和收敛性。