SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 334-369, January 2021.
Proper orthogonal decomposition (POD) stabilized methods for the Navier--Stokes equations are considered and analyzed. We consider two cases: the case in which the snapshots are based on a non inf-sup stable method and the case in which the snapshots are based on an inf-sup stable method. For both cases we construct approximations to the velocity and the pressure. For the first case, we analyze a method in which the snapshots are based on a stabilized scheme with equal order polynomials for the velocity and the pressure with local projection stabilization (LPS) for the gradient of the velocity and the pressure. For the POD method we add the same kind of LPS stabilization for the gradient of the velocity and the pressure as the direct method, together with grad-div stabilization. In the second case, the snapshots are based on an inf-sup stable Galerkin method with grad-div stabilization and for the POD model we also apply grad-div stabilization. In this case, since the snapshots are discretely divergence-free, the pressure can be removed from the formulation of the POD approximation to the velocity. To approximate the pressure, needed in many engineering applications, we use a supremizer pressure recovery method. Error bounds with constants independent of inverse powers of the viscosity parameter are proved for both methods. Numerical experiments show the accuracy and performance of the schemes.

中文翻译：

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不可压缩流正确正交分解稳定化方法的误差分析

SIAM数值分析学报，第59卷，第1期，第334-369页，2021年1月。
考虑并分析了Navier-Stokes方程的正确正交分解（POD）稳定方法。我们考虑两种情况：快照基于非infsup稳定方法的情况和快照基于inf-sup稳定方法的情况。对于这两种情况，我们都构造了速度和压力的近似值。对于第一种情况，我们分析了一种方法，其中快照基于稳定方案，该稳定方案对速度和压力使用等阶多项式，对速度和压力的梯度使用局部投影稳定（LPS）。对于POD方法，我们为速度和压力的梯度添加了与直接方法相同的LPS稳定度，以及grad-div稳定度。在第二种情况下 快照基于具有grad-div稳定的inf-sup稳定Galerkin方法，对于POD模型，我们也应用grad-div稳定。在这种情况下，由于快照没有离散离散，因此可以从POD近似于速度的公式中消除压力。为了估算许多工程应用中所需的压力，我们使用了超压回收方法。两种方法都证明了常数的误差范围与粘度参数的反幂无关。数值实验表明了该方案的准确性和性能。压力可以从POD近似于速度的公式中删除。为了估算许多工程应用中所需的压力，我们使用了超压回收方法。两种方法都证明了常数的误差范围与粘度参数的反幂无关。数值实验表明了该方案的准确性和性能。压力可以从POD近似于速度的公式中删除。为了估算许多工程应用中所需的压力，我们使用了超压回收方法。两种方法都证明了常数的误差范围与粘度参数的反幂无关。数值实验表明了该方案的准确性和性能。