Error analysis for a Crouzeix-Raviart approximation of the variable exponent Dirichlet problem

Abstract: In the present paper, we examine a Crouzeix-Raviart approximation of the $p(\cdot)$-Dirichlet problem. We derive a $\textit{medius}$ error estimate, $\textit{i.e.}$, a best-approximation result, which holds for uniformly continuous exponents and implies $\textit{a priori}$ error estimates, which apply for H\"older continuous exponents and are optimal for Lipschitz continuous exponents. Numerical experiments are carried out to review the theoretical findings.