Two spectral methods for 2D quasi-periodic scattering problems

We consider the 2D quasi-periodic scattering problem in optics, which has been modelled by a boundary value problem governed by Helmholtz equation with transparent boundary conditions. A spectral collocation method and a tensor product spectral method are proposed to numerically solve the problem on rectangles. The discretization parameters can be adaptively chosen so that the numerical solution approximates the exact solution to a high accuracy. Our methods also apply to solve general partial differential equations in two space dimensions, one of which is periodic. Numerical examples are presented to illustrate the accuracy and efficiency of our methods.