Analysis of a bilinear finite element for shallow shells I: Approximation of inextensional deformations

We consider a bilinear reduced-strain finite element formulation for a shallow shell model of Reissner-Naghdi type. The formulation is closely related to the facet models used in engineering practice. We estimate the error of this scheme when approximating an inextensional displacement field. We make the strong assumptions that the domain and the finite element mesh are rectangular and that the boundary conditions are periodic and the mesh uniform in one of the coordinate directions. We prove then that for sufficiently smooth fields, the convergence rate in the energy norm is of optimal order uniformly with respect to the shell thickness. In case of elliptic shell geometry the error bound is furthermore quasioptimal, whereas in parabolic and hyperbolic geometries slightly enhanced smoothness is required, except for the degenerate cases where the characteristic lines are parallel with the mesh lines. The error bound is shown to be sharp.