  We consider the Camassa-Holm (CH) equation, a nonlinear dispersive wave
equation that models one-way propagation of long waves of moderately small
amplitude. We discretize in space the periodic initial-value problem for CH
(written in its original and in system form), using the standard Galerkin
finite element method with smooth splines on a uniform mesh, and prove
optimal-order $L^{2}$-error estimates for the semidiscrete approximation. We
also consider an initial-boundary-value problem on a finite interval for the
system form of CH and analyze the convergence of its standard Galerkin
semidiscretization. Using the fourth-order accurate, explicit, "classical"
Runge-Kutta scheme for time-stepping, we construct a highly accurate, stable,
fully discrete scheme that we employ in numerical experiments to approximate
solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the
`peakon' type.
