Grégoire Pourtier
I am a PhD student in the research group for analysis and applications at TU Berlin. My research focuses on the numerical solution of scattering problems, primarily using boundary integral equation methods.
Intervention
The Helmholtz equation is essential to modeling wave scattering problems in periodic structures. Green’s functions, which solve linear partial differential equations with Dirac delta function sources δ(x), provide a foundation for recasting these problems into boundary integral equations. While closed-form expressions exist for many settings, the quasi-periodic Green’s function for the Helmholtz equation poses a unique challenge: it is defined as an infinite series that converges slowly as x2 → 0 in 2D (or x3 → 0 in 3D).
In this talk, we introduce QPGreen.jl, which implements an FFT-based algorithm to efficiently evaluate these Green’s functions. It combines spectral truncation and interpolation to overcome the limitations of direct series summation. Leveraging Julia’s composability, we combine QPGreen.jl with Inti.jl—a boundary integral equation solver—and demonstrate applications to wave scattering problems with quasi-periodic incident fields.