Energy estimates in hierarchical plate theories

The starting assumption of hierarchical plate theories is a representation of the displacement field in the form u((n))(x, zeta) = Sigma(k=0)(n) phi(k=0)(zeta)b(k)((n))(x), where x stands for the in-plane coordinates and zeta for the transverse coordinate, where the linearly independent functions phi(k) are the first (n + 1) in a complete system, and where the functions b(k)((n)) are determined by solving a minimum problem in an approximation space V-(n). For V the function space where the exact solution u is sought to the three-dimensional problem that a given hierarchical plate theory is meant to approximate, we show that the sequence {u((n))} converges in energy to a limit element u((infinity)) is an element of V, whatever the functions phi(k); and that, if phi(k)(zeta) = zeta(k), then u and u((infinity)) coincide pointwise, provided their difference is smooth, (C) 2000 Academic Press.