A multi-domain spectral-Galerkin method for the Neumann problem on quadrilaterals

A multi-domain spectral-Galerkin method for the Neumann problem on quadrilaterals is proposed. We establish first some results on the composite approximation of Legendre in Sobolev spaces with Jacobi weights. Then a multi-domain a multi-domain spectral-Galerkin scheme is provided for the elliptic equations with the Neumann boundary conditions. Efficient algorithms are presented. The accuracy of the numerical results agrees well with the theoretical analysis. Additionally, the algorithms are efficient for oscillation solutions with the boundary conditions of Neumann on quadrilaterals.