Upper and lower error bounds for plate-bending finite elements

We compare the robustness of three different low-order mixed methods that have been proposed for plate-bending problems: the so-called MITC, Arnold-Falk and Arnold-Brezzi elements. We show that for free plates, the asymptotic rate of convergence in the presence of quasiuniform meshes approaches the optimal O(h) for MITC elements as the thickness approaches 0, but only approaches O(h(1/2)) for the latter two. We accomplish this by establishing lower bounds for the error in the rotation. The deterioration occurs due to a consistency error associated with the boundary layer - we show how a modification of the elements at the boundary can fix the problem. Finally, we show that the Arnold-Brezzi element requires extra regularity for the convergence of the limiting (discrete Kirchhoff) case, and show that it fails to converge in the presence of point loads.