A priori and a posteriori error analysis of the Crouzeix-Raviart and Morley FEM with original and modified righthand sides

TL;DR

This paper analyzes the error properties of nonconforming finite element methods, specifically Crouzeix-Raviart and Morley elements, for harmonic problems with original and modified right-hand sides, providing a posteriori error estimates and convergence insights.

Contribution

It introduces a new analysis framework for nonconforming FEM with modified right-hand sides, including convergence rates and a posteriori error estimates with explicit constants.

Findings

Modified schemes recover best-approximation properties.

Original methods may outperform modified schemes with oscillating data.

Explicit reliable error estimates are provided for practical error control.

Abstract

This article on nonconforming schemes for $m$ harmonic problems simultaneously treats the Crouzeix-Raviart ($m=1$) and the Morley finite elements ($m=2$) for the original and for modified right-hand side $F$ in the dual space $V^*:=H^{-m}(\Omega)$ to the energy space $V:=H^{m}_0(\Omega)$. The smoother $J:V_{\rm nc} \to V$ in this paper is a companion operator, that is a linear and bounded right-inverse to the nonconforming interpolation operator $I_{\rm nc} :V\to V_{\rm nc}$, and modifies the discrete right-hand side $F_h:=F\circ J \in V_{\rm nc}^*$. The best-approximation property of the modified scheme from Veeser et al. (2018) is recovered and complemented with an analysis of the convergence rates in weaker Sobolev norms. Examples with oscillating data show that the original method may fail to enjoy the best-approximation property but can also be better than the modified scheme. The a~posteriori analysis of this paper concerns data oscillations of various types in a class of right-hand sides $F\in V^*$. The reliable error estimates involve explicit constants and can be recommended for explicit error control of the piecewise energy norm. The efficiency follows solely up to data oscillations and examples illustrate this can be problematic.