Upper and lower error bounds for
plate-bending finite elements
Juhani Pitkaranta and Manil Suri
juhani.pitkaranta@hut.fi,
suri@math.umbc.edu
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 $O(h^{\frac{1}{2}})$ for the
latter two. We accomplish this by establishing
{\em 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.
