A Convergent Crank-Nicolson Galerkin Scheme for the Benjamin-Ono Equation

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 $H^{1\mathrm{/}2}$-norm of the approximations locally. This enables us to show that the scheme converges strongly in $L^2(0,T;L_{\text{loc}}^2(\mathbb{R}))$ 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.