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

TL;DR

We address the numerical approximation of the nonlinear $p(\cdot)$-Dirichlet problem with Dirichlet boundary data using the nonconforming Crouzeix–Raviart element. The authors develop a medius error analysis for variable exponents, leveraging a node-averaging quasi-interpolation and local efficiency via shifted $N$-functions, and derive a priori error estimates that are optimal for Hölder exponents and quasi-optimal for Lipschitz exponents (with $\delta>0$). They also establish discrete duality and a Marini-type flux reconstruction, and validate the theory through numerical experiments. The results extend the error-control toolkit for variable-exponent PDEs and nonconforming FEM, providing rigorous guarantees for simulations in contexts with energy-gap phenomena and spatially varying growth.

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 Hölder continuous exponents and are optimal for Lipschitz continuous exponents. Numerical experiments are carried out to review the theoretical findings.

Theorems & Definitions (44)

• Remark 1.4
• Remark 2.6
• Proposition 2.9
• proof
• Lemma 2.14
• proof
• Remark 2.17: Natural distance
• Remark 2.18: Conjugate natural distance
• Remark 2.31
• Theorem 2.1: Best-approximation