Fourier embedded domain methods: Periodic and C∞ extension of a function defined on an irregular region to a rectangle via convolution with Gaussian kernels

One possible way to solve a partial differential equation in an irregular region Ω is the use of the so-called domain embedding methods, in where the domain of interest is embedded within a rectangle. In order to apply a Fourier spectral method on the rectangle, the inhomogeneous term f(x, y) has to be extended to a new function g(x, y) that is periodic and infinitely differentiable, and equal to f(x, y) everywhere within Ω. Some authors have given explicit methods to compute extensions with infinite order convergence for the cases in where the boundary of Ω, ∂Ω, can be defined as the zero isoline of a function ψ(x, y). For the cases in where this is not possible, we suggest a new method to build these extensions via convolution with Gaussian kernels.