A Lower Bound of the $L^2$ Norm Error Estimate for the Adini Element of the Biharmonic Equation

This paper is devoted to the $L^2$ norm error estimate of theAdini element for the biharmonic equation. Surprisingly, a lower bound isestablished which proves that the $L^2$ norm convergence ratecannot be higher than that in the energy norm.This proves the conjecture of [Lascaux and Lesaint, RAIRO Anal. Numer. , 9 (1975), pp. 9--53] that the convergence rates in both $L^2$ and $H^1$ normscannot be higher than that in the energy norm for this element.