论文标题
对不可压缩流的正交分解稳定方法的错误分析
Error analysis of proper orthogonal decomposition stabilized methods for incompressible flows
论文作者
论文摘要
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，即快照基于非Inf-Sup稳定方法以及快照基于Inf-Sup稳定方法的情况。对于这两种情况，我们构建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，该方法基于稳定的方案，该方案具有相等的速度多项式的稳定方案，以及速度和压力梯度的局部投影稳定（LPS）的压力。对于POD方法，我们为速度的梯度和压力添加了相同类型的LPS稳定，而不是直接方法，以及Grad-Div稳定。在第二种情况下，快照是基于具有Grad-Div稳定化的Inf-Sup稳定Galerkin方法，对于POD模型，我们还采用了毕业划线稳定。在这种情况下，由于快照是无分解的，因此可以从POD近似的配方中去除压力到速度。为了近似于许多工程应用所需的压力，我们使用了Aupremizer压力回收法。对于两种方法，都证明了与常数独立于粘度参数反向功率的误差界限。数值实验显示了方案的准确性和性能。
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 than 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 apply also 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 on inverse powers of the viscosity parameter are proved for both methods. Numerical experiments show the accuracy and performance of the schemes.
