Convergence Rates of a Fully Discrete Galerkin Scheme for the Benjamin-Ono Equation

We consider a recently proposed fully discrete Galerkin scheme for the Benjamin-Ono equation which has been found to be locally convergent in finite time for initial data in $L^2(\mathbb{R})$. By assuming that the initial data is sufficiently regular we obtain theoretical convergence rates for the scheme both in the full line and periodic versions of the associated initial value problem. These rates are illustrated with some numerical examples.