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 membraneand 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−1h1+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(t2) as t→ 0. In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound O(h).