The lower bound of the error estimate in the L2 norm for the Adini element of the biharmonic equation

TL;DR

This paper establishes a lower bound for the L2 norm error of the Adini element in solving the biharmonic equation, confirming that its convergence rate cannot surpass that in the energy norm, thus settling a long-standing conjecture.

Contribution

It proves that the L2 norm convergence rate of the Adini element is limited by the energy norm rate, confirming a conjecture from previous research.

Findings

L2 norm error convergence rate cannot exceed energy norm rate

Confirms the conjecture by Lascaux and Lesaint from 1975

Provides theoretical lower bounds for the Adini element's error estimates

Abstract

This paper is devoted to the $L^2$ norm error estimate of the Adini element for the biharmonic equation. Surprisingly, a lower bound is established which proves that the $ L^2$ norm convergence rate can not be higher than that in the energy norm. This proves the conjecture of [Lascaux and Lesaint, Some nonconforming finite elements for the plate bending problem, RAIRO Anal. Numer. 9 (1975), pp. 9--53.] that the convergence rates in both $L^2$ and $H^1$ norms can not be higher than that in the energy norm for this element.