Spectral methods for partial differential equations in irregular domains:: The spectral smoothed boundary method

In this paper, we propose a numerical method to approximate the solution of partial differential equations in irregular domains with no-flux boundary conditions. The idea is to embed the domain into a box and use a smoothing term to encode the boundary conditions into a modified equation that can be approached by standard spectral methods. The main features of this method are its capability to deal with domains of arbitrary shape and its easy implementation via fast Fourier transform routines. We discuss several examples of practical interest and test the results against standard numerical methods.