  First-order convergence in time and space is proved for a fully discrete
semi-implicit finite element method for the two-dimensional Navier--Stokes
equations with $L^2$ initial data in convex polygonal domains, without extra
regularity assumptions or grid-ratio conditions. The proof utilises the
smoothing properties of the Navier--Stokes equations, an appropriate duality
argument, 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. Numerical
examples are provided to support the theoretical analysis.
