Numerical assessment of a class of uniformly stable mixed spectral elements for the Navier–Stokes equations

In 1999, Bernardi and Maday analyzed a new class of mixed spectral elements for the Stokes and the Navier–Stokes equations [Bernardi C, Maday Y. Uniform Inf–Sup condition for the spectral discretization of the Stokes problem. Math Models Meth Appl Sci 1999;3:395–414] where they proved some interesting results like the uniform Inf–Sup condition. The main advantage we see is that applying the Uzawa algorithm to the discrete Stokes system yields a well-conditioned problem on the pressure. Then, the mass matrix preconditioned Conjugate Gradient method PCG used to compute the pressure converges in a number of iterations that is independent of the polynomial degree approximation. This paper presents the “numerical proofs” of the theoretical predictions on the stability and the accuracy of these spectral methods in mono-domain and multi-domain configurations.