Legendre spectral methods for the − grad(div) operator

Abstract

This paper describes two Legendre spectral methods for the − grad(div) eigenvalue problem in R 2 . The first method uses a single grid resulting from the P N ⊗ P N discretization in primal and dual variational formulations. As is well-known, this method is unstable and exhibits spectral ‘pollution’ effects: increased number of singular eigenvalues, and increased multiplicity of some eigenvalues belonging to the regular spectrum. Our study aims at the understanding of these effects. The second spectral method is based on a staggered grid of the P N ⊗ P N - 1 discretization. This discretization leads to a stable algorithm, free of spurious eigenmodes and with spectral convergence of the regular eigenvalues/eigenvectors towards their analytical values. In addition, divergence-free vector fields with sufficient regularity properties are spectrally projected onto the discrete kernel of − grad(div), a clear indication of the robustness of this algorithm.