朝着可计算的流和稳健估计的方向：应用于时间依赖不可压缩纳维-斯托克斯方程的inf- sup稳定的有限元方法
Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations
AI总结：
本文研究了应用于时间依赖不可压缩纳维-斯托克斯流的inf- sup稳定的有限元方法，重点在于动能和耗散能量的稳健估计，提出了压力鲁棒性和Re半鲁棒性两种方法，并讨论了正则性假设。
摘要：
考虑应用于时间依赖不可压缩纳维-斯托克斯流的inf- sup稳定的有限元方法。重点在于动能和耗散能量的稳健估计，从两个方面来看。首先，压力鲁棒性确保了基本不变性原理的满足，速度误差估计不受压力近似性的影响。其次，Re半鲁棒性意味着出现在动能和耗散能量误差估计右边的常数（包括格龙瓦尔常数）不显式依赖于雷诺数。这些估计依赖于基本正则性假设∇u ∈ L¹(0,T;L∞(Ω))，该假设进行了详细讨论。从最佳实践的角度，我们回顾并建立了对于点wise无散H¹符合有限元（如斯科特-沃格尔ius配对或某些等几何基础有限元）和点wise无散H(div)符合不连续伽辽金有限元的鲁棒和Re半鲁棒估计。对于对流占优问题，后者自然包含一个非梯度的迎风稳定化，用于速度。
英文摘要：
Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, $Re$-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption $\nabla u \in L^1(0,T;L^\infty(Ω))$ which is discussed in detail. In the sense of best practice, we review and establish pressure- and $Re$-semi-robust estimates for pointwise divergence-free $H^1$-conforming FEM (like Scott-Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free $H$(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.
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