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

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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.