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.