A SPECTRAL METHOD FOR THE EIGENVALUE PROBLEM FOR ELLIPTIC EQUATIONS

Let Omega be an open, simply connected, and bounded region in R-d, d >= 2, and assume its boundary partial derivative 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].