  In this paper we study the convergence of a fully discrete Crank-Nicolson
Galerkin scheme for the initial value problem associated with the fractional
Korteweg-de Vries (KdV) equation, which involves the fractional Laplacian and
non-linear convection terms. Our proof relies on the Kato type local smoothing
effect to estimate the localized $H^{\alpha/2}$-norm of the approximated
solution, where $\alpha \in [1,2)$. We demonstrate that the scheme converges
strongly in $L^2(0,T;L^2_{loc}(\mathbb{R}))$ to a weak solution of the
fractional KdV equation provided the initial data in $L^2(\mathbb{R})$.
Assuming the initial data is sufficiently regular, we obtain the rate of
convergence for the numerical scheme. Finally, the theoretical convergence
rates are justified numerically through various numerical illustrations.
