Selections of shape functions for dimensional reduction to Helmholtz's equation

The boundary value problem of Helmholtz's equation on a n + 1 dimensional thin slab is approximated by appropriate systems of the n-dimensional boundary value problem. The very detailed estimates for modeling error in the H¹-norm demonstrate convergence when the thickness of the slab approaches 0 as well as when the size of the systems approaches infinity. Shape functions through the thickness are first selected by finitely many eigenfunctions, and the tail is then selected to consist of polynomials. The presence of two types of functions gives rise to a certain choice in the selection of a particular set of shape functions. Numerical results provide a good illustration of the effect of different choices for specific problems.