Helmholtz equation in periodic media with a line defect

We consider the Helmholtz equation in an unbounded periodic media perturbed by an unbounded defect whose structure is compatible with the periodicity of the underlying media. We exhibit a method coupling Dirichlet-to-Neumann maps with the Lippmann–Schwinger equation approach to solve this problem, where the Floquet–Bloch transform in the direction of the defect plays a central role. We establish full convergence estimates that makes the link between the rate of decay of a function and the good behavior of a quadrature rule to approximate the inverse Floquet–Bloch transform. Finally we exhibit a few numerical results to illustrate the efficiency of the method.