  In this paper we prove the convergence of a Crank-Nicolson type Galerkin
finite element scheme for the initial value problem associated to the
Benjamin-Ono equation. The proof is based on a recent result for a similar
discrete scheme for the Korteweg-de Vries equation and utilizes a local
smoothing effect to bound the $H^{1/2}$-norm of the approximations locally.
This enables us to show that the scheme converges strongly in
$L^{2}(0,T;L^{2}_{\text{loc}}(\mathbb{R}))$ to a weak solution of the equation
for initial data in $L^{2}(\mathbb{R})$ and some $T > 0$. Finally we illustrate
the method with some numerical examples.
