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Interior penalty discontinuous Galerkin FEM for the $p(x)$-Laplacian

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 $p:Ω\to [p_1,p_2]$ is log Hölder 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.