Finite element approximation of the $p(\cdot )$-Laplacian

Abstract

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 $\alpha$-Hölder continuous. The error of the gradients is measured in the so-called quasi-norm, i.e. we measure the $L^2$-error of $|\nabla v{|}^{\frac{p-2}{2}}\nabla v$.

Cited by in corpus (2)

• Order of convergence of the finite element method for the $p(x)-$Laplacian
• The decomposition-coordination method for the $p(x)$-Laplacian