Two spectral methods for 2D quasi-periodic scattering problems

TL;DR

This paper introduces two spectral methods for accurately and efficiently solving 2D quasi-periodic scattering problems in optics, with adaptive discretization and broad applicability to PDEs.

Contribution

The paper proposes a spectral collocation and a tensor product spectral method for 2D quasi-periodic scattering problems, enhancing accuracy and efficiency with adaptive parameter selection.

Findings

High-accuracy solutions demonstrated through numerical examples

Methods effectively handle general 2D PDEs with periodicity

Adaptive discretization improves computational efficiency

Abstract

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.