Error analysis of Legendre spectral method with essential imposition of Neumann boundary condition

In this paper, we present error estimates of Legendre spectral method with essential imposition of Neumann boundary condition. The algorithm was firstly proposed by Auteri, Parolini and Quartapelle. This method differs from the classical spectral methods for Neumann boundary value problems. The homogeneous boundary condition is satisfied exactly. Moreover, a double diagonalization process is employed, instead of the full stiffness matrices encountered in the classical variational formulation of the problem with a weak natural imposition of the derivative boundary condition. We also consider nonhomogeneous Neumann data by means of a lifting. In particular, the lifting in this paper is expressed explicitly and is different from that by Auteri, Parolini and Quartapelle. For analyzing the numerical errors, some basic results on Legendre quasi-orthogonal approximations are established. The convergence of proposed schemes is proved.