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

TL;DR

This paper develops an interior penalty discontinuous Galerkin method to approximate solutions of the $p(x)$-Laplacian variational problem, proving convergence and demonstrating its effectiveness through numerical experiments relevant to image processing.

Contribution

It introduces a novel interior penalty discontinuous Galerkin approach for the $p(x)$-Laplacian, with convergence analysis and comparative numerical results.

Findings

Minimizers of the discrete functional converge to the continuous solution.

Numerical experiments show the method's effectiveness in one-dimensional cases.

Comparison indicates advantages over conforming Galerkin methods when $p_1$ is close to one.

Abstract

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.