Finite Element Approximation of the $p(\cdot)$-Laplacian
Dominic Breit, Lars Diening, Sebastian Schwarzacher · SIAM Journal on Numerical Analysis · 2015
We study a priori estimates for the $p(\cdot)$-Laplace Dirichlet problem, $-\mathrm{div}(\vert{ abla \mathbf{v}}\vert^{p(\cdot)-2} abla \mathbf{v}) = \mathbf{f}$. 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 $\alpha$-Hölder continuous. The error of the gradients is measured in the so-called quasi-norm, i.e., we measure the $L^2$-error of $\vert{ abla \mathbf{v}}\vert^{\frac{p-2}{2}} abla \mathbf{v}$.