Energy estimates in hierarchical plate theories (Q1580474)

The starting assumption of hierarchical plate theories is a representation of displacement field in the form \({\mathbf u}^{(n)} (x,\xi)=\sum_{k=0}^n\varphi_k(\xi){\mathbf b}^{(n)}_k(x)\), where \(x\) stands for the in-plane coordinates and \(\xi\) for the transverse coordinate. The linearly independent functions \(\varphi_k\) are the first \((n+1)\) in a complete system, and the function \({\mathbf b}_k^{(n)}\) are determined by solving a minimum problem in an approximation space \({\mathcal V}^{(n)}\). For the function space \({\mathcal V}\) where the exact solution \textbf{u} is sought, the authors show that the sequence \(\{{\mathbf u}^{(n)}\}\) converges in energy norm to a limit element \({\mathbf u}^{(\infty)}\in {\mathcal V}\), whatever the functions \(\varphi_k\) are, and that, if \(\varphi_k(\xi)=\xi^k\), then \textbf{u} and \({\mathbf u}^{(\infty)}\) coincide pointwise, provided their difference is smooth.