A Spectral Method for the Eigenvalue Problem for Elliptic Equations
For researchers in numerical PDEs, this is an incremental extension of existing spectral methods.
The paper proposes a spectral method for numerically solving eigenvalue problems for elliptic PDEs over smooth domains, extending prior work. Numerical illustrations demonstrate the method's effectiveness.
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].