We consider a sequence of linear Dirichlet problems as follows $$\begin{cases}-\dive ( \s_\e \nabla u_\e) = f \; \text{in }\, \O, \cr u_\e \in H^1_0(\O),\end{cases} $$ with $(\s_\e)$ uniformly elliptic and possibly non-symmetric.
Using \emph{purely variational arguments} we give an alternative proof of the compactness of $H$-convergence, originally proved by Murat and Tartar.
