A Spectral Method for the Eigenvalue Problem for Elliptic Equations

Let $\mathrm{\Omega }$ be an open, simply connected, and bounded region in ${\mathbb{R}}^d$, $d\ge 2$, and assume its boundary $\mathrm{\partial }\mathrm{\Omega }$ is smooth. Consider solving the eigenvalue problem $Lu=\lambda u$ for an elliptic partial differential operator $L$ over $\mathrm{\Omega }$ 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].