About the ellipticity of the Chebyshev–Gauss–Radau discrete Laplacian with Neumann condition

The Chebyshev–Gauss–Radau discrete version of the polar-diffusion operator, 1 r ∂ ∂ r r ∂ ∂ r - k 2 r 2 , k being the azimuthal wave number, presents complex conjugate eigenvalues, with negative real parts, when it is associated with a Neumann boundary condition imposed at r = 1. It is shown that this ellipticity marginal violation of the original continuous problem is genuine and not due to some round-off error amplification. This situation, which does not lead per se to any particular computational difficulty, is taken here as an opportunity to numerically check the sensitivity of the quoted ellipticity to slight changes in the mesh. A particular mapping is chosen for that purpose. The impact of this option on the ellipticity and on the numerical accuracy of a computed flow is evaluated.