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We consider the 2D quasi-periodic scattering problem in optics, which hasbeen modelled by a boundary value problem governed by Helmholtz equation withtransparent boundary conditions. A spectral collocation method and a tensorproduct spectral method are proposed to numerically solve the problem onrectangles. The discretization parameters can be adaptively chosen so that thenumerical solution approximates the exact solution to a high accuracy. Ourmethods also apply to solve general partial differential equations in two spacedimensions, one of which is periodic. Numerical examples are presented toillustrate the accuracy and efficiency of our methods.

Two spectral methods for 2D quasi-periodic scattering problems