压力诱导的混合方法在时间依赖的（纳维-斯托克斯）方程中的锁定现象
Pressure-induced locking in mixed methods for time-dependent (Navier-)Stokes equations
AI总结：
本文研究了时间依赖的不可压缩斯托克斯和纳维-斯托克斯方程的稳定混合方法，发现经典稳定混合方法在动量平衡受梯度力主导时会出现锁定现象，而结构保持的混合方法则不受影响。
摘要：
我们考虑了时间依赖的不可压缩斯托克斯和纳维-斯托克斯方程的稳定混合方法，扩展了之前关于稳态（纳维-）斯托克斯问题的研究。通过一种新颖、优雅且简单的数值分析和相应的数值实验，发现经典稳定混合方法如泰勒-霍德或克鲁兹-拉维尔特元素在动量平衡受梯度力主导时会出现锁定现象。更具体地说，分析预测了高阶方法的L²收敛阶降低，甚至在预渐近网格上最低阶方法的L²收敛阶完全停滞，并在实践中观察到。另一方面，也证明了（结构保持）压力鲁棒混合方法不受此锁定现象影响，即使它们是最低阶的。还建立了与（矢量）超几何守恒律的well-balanced方案（如浅水或可压缩欧拉方程）之间的联系。
英文摘要：
We consider inf-sup stable mixed methods for the time-dependent incompressible Stokes and Navier--Stokes equations, extending earlier work on the steady (Navier-)Stokes Problem. A locking phenomenon is identified for classical inf-sup stable methods like the Taylor-Hood or the Crouzeix-Raviart elements by a novel, elegant and simple numerical analysis and corresponding numerical experiments, whenever the momentum balance is dominated by forces of a gradient type. More precisely, a reduction of the $L^2$ convergence order for high order methods, and even a complete stall of the $L^2$ convergence order for lowest-order methods on preasymptotic meshes is predicted by the analysis and practically observed. On the other hand, it is also shown that (structure-preserving) pressure-robust mixed methods do not suffer from this locking phenomenon, even if they are of lowest-order. A connection to well-balanced schemes for (vectorial) hyperbolic conservation laws like the shallow water or the compressible Euler equations is made.
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