Legendre spectral methods for the -grad (div) operator with free boundary conditions

Abstract

This paper extends previous studies of the application of Legendre spectral methods to the grad (div) eigenvalue problem on a quadrangular domain in $I\!\!R^2$. 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. $I\!\!P_N \otimes I\!\!P_N$) and staggered (i.e. $I\!\!P_N \otimes I\!\!P_{N-1}$) 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.