The Best $L^2$ Norm Error Estimate of Lower Order Finite Element Methods for the Fourth Order Problem (Journal of Computational Mathematics, 2018)
In the paper, we analyze the $L^2$ norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the $L^2$ norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including: the Powell-Sabin $C^1-P_2$ macro element, the nonconforming Morley element, the $C^1-Q_2$ macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming Wang-Xu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature: can the $L^2$ norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm?