A spectral method for the eigenvalue problem for elliptic equations

Kendall Atkinson and Olaf Hansen

Abstract

Let $\Omega$ be an open, simply connected, and bounded region in ${\bf R}^{d}$, $d\geq2$, 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 by Atkinson, Chien, and Hansen [Adv. Comput. Math, 33 (2010), pp. 169–189, and to appear].

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Key words

elliptic equations, eigenvalue problem, spectral method, multivariable approximation

65M70

Links to the cited ETNA articles

 [7] Vol. 17 (2004), pp. 206-217 Kendall Atkinson and Weimin Han: On the numerical solution of some semilinear elliptic problems