Fourier embedded domain methods: extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and C∞

A simple way to solve a partial differential equation in a non-rectangular domain Ω is to embed the domain in a rectangle B and solve the problem (more easily) in the rectangle. To apply a Fourier spectral method on the rectangle, it is necessary to extend the inhomogeneous term in the PDE, f(x,y), to B in such a way that the extended function g(x,y) is periodic and infinitely differentiable, and yet is equal to f(x,y) everywhere in the irregular domain Ω. If the boundary of Ω, ∂Ω, is defined as the zero isoline of a function Φ(x,y), then a suitable extension is g(x,y)≡f(x,y)H(1−2Φ(x,y)/constant) where H is a smoothed approximation to the step function which is defined in the main text.