Finite Element Approximation of the p(·)-Laplacian

We study a priori estimates for the $p(\cdot)$-Laplace Dirichlet problem, $-\mathrm{div}(\vert{\nabla \mathbf{v}}\vert^{p(\cdot)-2} \nabla \mathbf{v}) = \mathbf{f}$. We show that the gradients of the finite element approximation with zero boundary data converge with rate $O(h^\alpha)$ if the exponent $p$ is $\alpha$-Holder continuous. The error of the gradients is measured in the so-called quasi-norm, i.e., we measure the $L^2$-error of $\vert{\nabla \mathbf{v}}\vert^{\frac{p-2}{2}} \nabla \mathbf{v}$.