This article concerns the numerical approximation of the two-dimensional nonstationary Navier–Stokes equations with \(H^1\) initial data. By utilizing special locally refined temporal stepsizes, we prove that the linearly extrapolated Crank–Nicolson scheme, with the usual stabilized Taylor–Hood finite element method in space, can achieve second-order convergence in time and space. Numerical examples are provided to support the theoretical analysis.

中文翻译：

---

具有 $$\mathbf{H^1}$$ H 1 初始数据的 Navier-Stokes 方程的线性外推 Crank-Nicolson 方法的二阶收敛

本文涉及具有\(H^1\)初始数据的二维非平稳 Navier-Stokes 方程的数值近似。通过利用特殊的局部细化时间步长，我们证明了线性外推的 Crank-Nicolson 方案，以及通常在空间中稳定的 Taylor-Hood 有限元方法，可以在时间和空间上实现二阶收敛。提供了数值例子来支持理论分析。