A Spectral Method for the Eigenvalue Problem for Elliptic Equations

Abstract

Let $Ω$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partialΩ$ is smooth. Consider solving the eigenvalue problem $Lu=λu$ for an elliptic partial differential operator $L$ over $Ω$ 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].