Second-Order Convergence of the Linearly Extrapolated Crank–Nicolson Method for the Navier–Stokes Equations with $$\mathbf{H^1}$$ Initial Data

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