Legendre spectral methods for the -grad (div) operator with free boundary conditions
Résumé
This paper extends previous studies of the application of Legendre spectral methods to the grad (div) eigenvalue problem on a quadrangular domain in. The extension focuses on natural boundary conditions. Spectral approximations based on primal and dual variational approaches are built using Gaussian quadrature rules both on single (i.e.) and staggered (i.e.) grids. The single grid approximation is unstable and exhibits 'spectral pollution' effects such as increased number of zero eigenvalues and increased multiplicity of some non-zero eigenvalues. The approximation on the staggered grid leads to a stable algorithm, free of spurious eigenmodes and with spectral convergence of the non-zero eigenvalues/eigenvectors towards their analytical values. © Springer Science + Business Media, LLC 2009.