Hierarchic modelling of separable elliptic boundary value problems on thin domains

The dimensional reduction method for solving Laplace's equation on a flat plate, an arch and a spherical shell is investigated, extending previous work on laminated plates by Vogelius and Babuska (1981). Convergence rates for the error in energy are obtained, extending previous results by deriving explicit values for the constant of approximation and its dependence on the thickness of the domain and the model order. The framework for laminated plates is shown to easily extend to other geometries. Numerical results are given which verify the convergence rates in terms of the thickness. Details are given as to how to implement the dimensional reduction technique and in particular, for a spherical shell, a method is given which reduces the problem to that of inverting relatively small matrices.;A posteriori error estimators are given for each of the geometries under consideration. Error estimators are already known for flat plates. It is shown how the estimators for flat plates can be modified for use in the arch case, and for shells, techniques for estimating the discretization error and modelling error are presented. The a posteriori estimators are then used to derive a refinement algorithm for adaptively constructing hierarchic models for representative problems on each of the geometries under consideration.