A Spectral Embedding Method Applied to the Advection Diffusion Equation

In order to solve partial differential equations in complex geometries with a spectral type method, one describes an embedding approach which essentially makes use of Fourier expansions and boundary integral equations. For the advection-diffusion equation, the method is based on an efficient "Helmholtz solver," the accuracy of which is tested by considering 1D and 2D Helmholtz-like problems. Finally, the capabilities of the method are pointed out by considering a 2D advection-diffusion problem in a hexagonal geometry.