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First-order convergence in time and space is proved for a fully discretesemi-implicit finite element method for the two-dimensional Navier--Stokesequations with $L^2$ initial data in convex polygonal domains, without extraregularity assumptions or grid-ratio conditions. The proof utilises thesmoothing properties of the Navier--Stokes equations, an appropriate dualityargument, and the smallness of the numerical solution in the discrete$L^2(0,t_m;H^1)$ norm when $t_m$ is smaller than some constant. Numericalexamples are provided to support the theoretical analysis.

Analysis of fully discrete finite element methods for 2D Navier--Stokes equations with critical initial data