We solve numerically the Serre-Green-Naghdi (SGN) system using stable, accurate and efficient fully discrete numerical schemes based on Galerkin/finite element methods. After reviewing the properties of the SGN system we present the convergence properties of the numerical scheme. A detailed study of the dynamics of the solitary waves of the SGN system over variable bottom topographies is also presented. It is noted that the Galerkin/finite element method is the only method analytically proven to be convergent for the numerical solution of the Serre equations so far.