Jacobi spectral method with essential imposition of Neumann boundary condition

In this paper, we propose Jacobi spectral method with essential imposition of Neumann boundary condition. This method differs from the classical spectral methods for Neumann boundary value problems. The homogeneous boundary condition is satisfied exactly. Moreover, a diagonal or tridiagonal matrix 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. For analyzing the numerical error, some basic results on Jacobi quasi-orthogonal and orthogonal approximations are established. The convergence of proposed schemes is proved. Numerical results demonstrate the efficiency of this approach and coincide well with theoretical analysis.