In this work, we study the convergence of the finite element method when applied to the following parabolic equation: $u_t = div (|u| ^ {\gamma (\mathbf{x})} \nabla u) + f (\mathbf{x}, t)$, $\mathbf{x}\in\Omega\subset R^{m}, t\in]0,T].$ Since the problem may be of degenerate type, we utilize an approximate problem, regularized by introducing a parameter $\varepsilon $. We prove, under certain conditions on $ \gamma $ and $ f $, that the weak solution of the approximate problem converges to the weak solution of the initial problem, when the parameter $ \varepsilon $ tends to zero. Discrete solutions are built using the finite element method and the convergence of these for the weak solution of the approximate problem is proved. Finally, we present some numerical results of a MATLAB implementation of the method.