  In this paper we construct an "Interior Penalty" Discontinuous Galerkin
method to approximate the minimizer of a variational problem related to the
$p(x)-$Laplacian. The function $p:\Omega\to [p_1,p_2]$ is log H\"{o}lder
continuous and $1<p_1\leq p_2<\infty$. We prove that the minimizers of the
discrete functional converge to the solution. We also make some numerical
experiments in dimension one to compare this method with the Conforming
Galerkin Method, in the case where $p_1$ is close to one. This example is
motivated by its applications to image processing.
