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Error analysis for a Crouzeix-Raviart approximation of the $p$-Dirichlet problem

Alex Kaltenbach

TL;DR

This work analyzes a Crouzeix–Raviart discretization for nonlinear $p$-Dirichlet problems with $(p,\delta)$-structure and establishes optimal $a$ priori error bounds for all $p\in(1,\infty)$ and $\delta\ge0$, along with medius (best-approximation) results and a reliable, efficient primal–dual a posteriori error estimator. By employing shifted $N$-functions and a node-averaging quasi-interpolation, the authors prove local efficiency and discrete duality results, enabling a comprehensive error analysis in the natural distance and its conjugate. A key contribution is extending medius analysis to general $p$, showing that CR and conforming Lagrange discretizations perform comparably under mild regularity assumptions, and deriving an a posteriori estimator that is provably reliable and efficient. Numerical experiments in a manufactured setting corroborate the theoretical findings and illustrate the expected convergence behavior and adaptivity gains. Overall, the paper provides a robust framework for error control in nonlinear CR approximations to the $p$-Dirichlet problem with $(p,\delta)$-structure, with implications for nonlinear PDE applications in continuum mechanics and related fields.

Abstract

In the present paper, we examine a Crouzeix-Raviart approximation for non-linear partial differential equations having a $(p,δ)$-structure for some $p\in (1,\infty)$ and $δ\ge 0$. We establish a priori error estimates, which are optimal for all $p\in (1,\infty)$ and $δ\ge 0$, medius error estimates, i.e., best-approximation results, and a primal-dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.

Error analysis for a Crouzeix-Raviart approximation of the $p$-Dirichlet problem

TL;DR

This work analyzes a Crouzeix–Raviart discretization for nonlinear -Dirichlet problems with -structure and establishes optimal priori error bounds for all and , along with medius (best-approximation) results and a reliable, efficient primal–dual a posteriori error estimator. By employing shifted -functions and a node-averaging quasi-interpolation, the authors prove local efficiency and discrete duality results, enabling a comprehensive error analysis in the natural distance and its conjugate. A key contribution is extending medius analysis to general , showing that CR and conforming Lagrange discretizations perform comparably under mild regularity assumptions, and deriving an a posteriori estimator that is provably reliable and efficient. Numerical experiments in a manufactured setting corroborate the theoretical findings and illustrate the expected convergence behavior and adaptivity gains. Overall, the paper provides a robust framework for error control in nonlinear CR approximations to the -Dirichlet problem with -structure, with implications for nonlinear PDE applications in continuum mechanics and related fields.

Abstract

In the present paper, we examine a Crouzeix-Raviart approximation for non-linear partial differential equations having a -structure for some and . We establish a priori error estimates, which are optimal for all and , medius error estimates, i.e., best-approximation results, and a primal-dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.
Paper Structure (31 sections, 25 theorems, 134 equations, 1 figure, 2 tables, 1 algorithm)

This paper contains 31 sections, 25 theorems, 134 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related contributions
  3. New contribution
  4. Outline
  5. Preliminaries
  6. Standard function spaces
  7. $N$-functions
  8. Basic properties of the non-linear operator
  9. Triangulations and standard finite element spaces
  10. Crouzeix--Raviart element
  11. Raviart--Thomas element
  12. Discrete integration-by-parts formula
  13. $p$-Dirichlet problem
  14. Variational problem
  15. Minimization problem and convex duality relations
  16. ...and 16 more sections

Key Result

Proposition 2.4

Let $\boldsymbol{\mathcal{A}}\colon\mathbb{R}^d \to \mathbb{R}^d$ satisfy Assumption assum:extra_stress for $p\in (1,\infty)$ and $\delta \ge 0$. Moreover, let $\varphi\colon\mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0}$ be defined by eq:def_phi and let $F,F^*\colon\mathbb{R}^d \to \mathbb{R}^d$ be defin The constants in eq:hammera and eq:hammerf depend only on the characteristics of ${\boldsymbol{\mat

Figures (1)

  • Figure 1: Plots of $\eta_k^2(v_k)$ and $\rho^2(v_k)$ for $p\in \{1.5,2,2.5,3\}$ and $v_k\coloneqq I_{h_k}^{\textit{av}}u_k^{\textit{cr}}\in{\mathcal{S}^1_D(\mathcal{T}_k)}$, using adaptive mesh refinement for $k=0,\dots,19$ and using uniform mesh refinement for $k=0,\dots,4$.

Theorems & Definitions (59)

  • Remark 2.1
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5: Change of shift
  • proof
  • Remark 2.6: Natural distance
  • Remark 2.7: Conjugate natural distance
  • Lemma 2.8
  • proof
  • ...and 49 more