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By the standard theory, the stable $Q_{k+1,k}$-$Q_{k,k+1}/Q_{k}^{dc'}$divergence-free element converges with the optimal order of approximation forthe Stokes equations, but only order $k$ for the velocity in $H^1$-norm and thepressure in $L^2$-norm. This is due to one polynomial degree less in $y$direction for the first component of velocity, a $Q_{k+1,k}$ polynomial. Inthis manuscript, we will show a superconvergence of the divergence free elementthat the order of convergence is truly $k+1$, for both velocity and pressure.Numerical tests are provided confirming the sharpness of the theory.

Superconvergence of the $Q_{k+1,k}$-$Q_{k,k+1}$ divergence-free finite element