P. Fischer and J. Mullen [3] have recently proposed a stabilization technique for the solution of the unsteady Navier-Stokes equations with the spectral element method. It is based on interpolations in physical space: Given the values of a function u on a Gauss-Lobatto-Legendre (GLL) mesh with (N + 1) d nodes per element (where d is the space dimension and N is the degree of the polynomial approximation in each direction), in each element one uses the polynomial interpolant to compute u at the N d nodes-GLL mesh, so that one obtains a new polynomial approximation, the degree of which in each direction is then N -1. Combined with a relaxation method, this "filtering" of the highest frequencies is applied at each time step. An important advantage of the technique is that interelement continuity and boundary conditions are preserved. Its efficiency, for high Reynolds number flows, was demonstrated from results of numerical simulations. In the present note we point out that this technique finds a simple interpretation in Legendre spectral space and that this interpretation remains true when a Chebyshev polynomial approximation is used. Although the result is easy to obtain it is not a priori obvious, so that to our knowledge this viewpoint has not yet been clearly stated. Moreover, we emphasize the link, only briefly mentioned in the conclusion of [3], to the filtering procedure suggested in [2], and finally we consider the case of the Fourier spectral method, by extension to the trigonometric polynomials.
Without loss of generality we can restrict ourselves to the one-dimensional situation. Let = [-1, 1], P N ( ), N ∈ N * , the space of the polynomials of maximum degree N defined on , {L i } N i=0 , the set of the N + 1 Legendre polynomials of degree i, and {ξ N i } N i=0 , the set of the GLL nodes associated with P N ( ), i.e., those that solve (1x 2 )L N (x) = 0. 646