Electron devices based on graphene have lately received a considerable interest; in fact, they could represent the ultimate miniaturization, since the active area is only one atom thick. However, the gapless dispersion relation of graphene at the Dirac points limits the possibility of using pristine graphene instead of traditional semiconductors in Field Effect Transistors (FET). For such a reason very accurate simulations are needed. In Nastasi and Romano (IEEE Trans. Electron Devices 68, 2021) a graphene field effect transistor (GFET) has been proposed and simulated adopting a drift-diffusion model. Here, electron devices whose active area is made of monolayer graphene are simulated taking as mathematical model the complete semiclassical Boltzmann transport equation (BTE) retaining the full complexity of the collision operator which includes all the electron-phonon scatterings along with the Pauli exclusion principle. The transport equation is coupled with the Poisson equation for the electric field. The Boltzmann equation is solved by means of a discontinuous Galerkin (DG) approach with linear elements in space and piecewise constant approximation in the wave-vector. The correct physical range for the distribution function is preserved with a maximum-principle-preserving scheme. Finite differences for the Poisson equation, and a TVD Runge-Kutta scheme for time discretization are adopted. The method reveals very robust and possesses a good degree of accuracy, making it particularly well suited for capturing the complex charge transport dynamics inherent to graphene-based devices. The kinetic approach reveals new features which are caught by the standard drift-diffusion equations; in particular, the presence of a boundary layer at the contacts.