Spectral methods are frequently used to replace a partial differential equations posed over an open set in R3by a seris of partial differential equations posed over an open set of lower dimesnsion and which are thus easier to apporximate. In the framework of three dimensional linearized elasticity those methods have already been used to derive two-dimensional approximate models for plates and one-dimensional approximate models for beams. The object of this paper is to show how to construct the "best" Galerkin spectral approximation for thin clamped plates, possibly multi-layered, made of homogeneous or of non-homogeneous materials. To this end, we use a technique introduced by M. Vogelius and I. Babu#x161;ka.
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