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

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 in the analysis of the consistency errors, 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.