  In this paper, we analyze and implement the Dirichlet spectral-Galerkin
method for approximating simply supported vibrating plate eigenvalues with
variable coefficients. This is a Galerkin approximation that uses the
approximation space that is the span of finitely many Dirichlet eigenfunctions
for the Laplacian. Convergence and error analysis for this method is presented
for two and three dimensions. Here we will assume that the domain has either a
smooth or Lipschitz boundary with no reentrant corners. An important component
of the error analysis is Weyl's law for the Dirichlet eigenvalues. Numerical
examples for computing the simply supported vibrating plate eigenvalues for the
unit disk and square are presented. In order to test the accuracy of the
approximation, we compare the spectral-Galerkin method to the separation of
variables for the unit disk. Whereas for the unit square we will numerically
test the convergence rate for a variable coefficient problem.
