The Fermi equation has diverse applications in different fields of science. This equation can be driven as an asymptotic limit of Boltzmann equation. This equation is degenerate in both convection and diffusion in the sense that drift and diffusion are taking place in, physically, different domains. Besides the problem is convection dominated since the diffusion term has a very small coefficient. The $h$ and $hp$ approximation of the streamline diffusion method and discontinuous Galerkin designed for the finite element analysis of this hyperbolic problem. We show that optimal convergence can be achieved subject to the regularity of the solution. In the second part of the current thesis, flow of rarified gas through a channel with arbitrary cross section which is governed by the Boltzmann equation is studied. Since the numerical effort for the numerical solution of Boltzmann equation is computationally involved, due to flow conditions, certain linearization of the governing kinetic equation are applied to reduce the number of space and velocity coordinates. The discrete velocity and streamline diffusion finite element method are combined to yield a numerical scheme. For this method, we derive stability and optimal error estimates in the $L_2$ norm. The optimality is due to maximal regularity of the exact solution of the corresponding hyperbolic pde. The potential of the proposed method is illustrated trough implementing some numerical examples.