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

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.