A spectral method for solving nonhomogeneous Neumann boundary value problems on quadrilaterals

In this paper, we develop a spectral method with essential imposition of nonhomogeneous Neumann boundary condition on quadrilaterals. Results on irrational orthogonal approximation are established, which facilitate the numerical solutions of partial differential equations of the sort. As applications, we provide spectral schemes for two model problems, and prove their spectral accuracy. Efficient numerical implementations are described. Numerical results demonstrate high efficiency of the suggested algorithms.