The least-squares pseudo-spectral method for Navier–Stokes equations

A spectral collocation approximation of first-order system least squares for incompressible Stokes equations was analyzed in Kim et al. (2004)  [12], and finite element approximations for incompressible Navier–Stokes equations were developed in Bochev et al. (1998,1999)  [9] and [10]. The aim of this paper is to analyze the first-order system least-squares pseudo-spectral method for incompressible Navier–Stokes equations. The paper will be an extension of the result in Kim et al. (2004)  [12] to the Navier–Stokes equations. Our least-squares functional is defined by the sum of discrete spectral norms of a first-order system of equations corresponding to the Navier–Stokes equations based on Legendre–Gauss–Lobatto points. We show its ellipticity and continuity over an appropriate product space, and spectral convergences of discretization errors are derived in the H1H1-norm and the L2L2-norm in each variable. Finally, we present some numerical examples.