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