Let $\Omega$ be an open, simply connected, and bounded region in
$\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth.
Consider solving the eigenvalue problem $Lu=\lambda u$ for an elliptic partial
differential operator $L$ over $\Omega$ with zero values for either Dirichlet
or Neumann boundary conditions. We propose, analyze, and illustrate a 'spectral
method' for solving numerically such an eigenvalue problem. This is an
extension of the methods presented earlier in [5],[6].