Analysis of a bilinear finite element for shallow shells. II: Consistency error

We consider a bilinear reduced-strain finite element of the MITC family for a shallow Reissner-Naghdi type shell. We estimate the consistency error of the element in both membrane- and bending-dominated states of deformation. We prove that in the membrane-dominated case, under severe assumptions on the domain, the finite element mesh and the regularity of the solution, an error bound $O(h + t^{-1}h^{1+s})$ can be obtained if the contribution of transverse shear is neglected. Here $t$ is the thickness of the shell, $h$ the mesh spacing, and $s$ a smoothness parameter. In the bending-dominated case, the uniformly optimal bound $O(h)$ is achievable but requires that membrane and transverse shear strains are of order $O(t^2)$ as $t \rightarrow 0$. In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound $O(h)$.