The kinetic energy of a flow is proportional to the square of the $L^2\left(\Omega \right)$ norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree $r$, then the best approximation error in $L^2\left(\Omega \right)$ is of order $r+1$. In this survey, the available finite element error analysis for the velocity error in $L^{\infty }(0,T;L^2\left(\Omega \right))$ is reviewed, where $T$ is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order $r-1$, $r$, and $r+1∕2$ for the velocity error in $L^{\infty }(0,T;L^2\left(\Omega \right))$. All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open.

流动的动能与流动的平方成正比 ${升}^2\left(\Omega \right)$速度的范数。给定足够的规则速度场和具有多项式的速度有限元空间$r$，那么最佳近似误差 ${升}^2\left(\Omega \right)$ 是有序的 $r+1$. 在本次调查中，可用有限元误差分析对速度误差${升}^{\infty }(0,吨;{升}^2\left(\Omega \right))$ 被审查，其中 $吨$是最后一次。由于在实践中特别关注小粘度系数或主要对流的情况，这可能导致湍流，因此考虑稳健的误差估计，即误差界限中的常数不依赖于粘度系数的反幂的估计. 可以推导出稳健估计的方法能够在相对粗糙的网格上对小粘度系数进行稳定的流动模拟，这在实践中经常遇到。为了引入对流主导机制的稳定技术和误差分析中使用的工具，首先研究了演化线性对流-扩散方程。本次调查的主要部分考虑了不可压缩的 Navier-Stokes 阶次方程的稳健有限元方法$r-1$, $r$， 和 $r+1∕2$ 对于速度误差 ${升}^{\infty }(0,吨;{升}^2\left(\Omega \right))$. 所有这些方法都进行了详细讨论。特别是，给出了误差界证明的草图，解释了最终确定收敛顺序的重要项的估计。其中，有用于 inf-sup 稳定有限元空间对以及压力稳定离散化的方法。数值研究支持其中几种方法的分析结果。此外，还调查了以稳健方式运行但只有非稳健误差分析可用的方法。本次调查的结论是，是否存在动能最优收敛阶次的鲁棒方法仍然是一个问题。