The spectral domain embedding method for partial differential equations on irregular domains

When the solution of a partial differential equation (PDE) is analytic in a regular computational domain, spectral methods are known to yield spectral convergence. However, standard spectral methods have great difficulties in handling a complex irregular computational domain $\Omega$ with boundary $\partial\Omega$.In the spectral domain embedding method, the irregular physical domain $\Omega$ is embedded into a rectangular computational domain $R$. This allows the application of spectral methods in the extended domain $R$ provided that the coefficient and the source terms can be extended smoothly from $\Omega$ to $R$.The rectangular domain $R$ is discretized with Chebyshev or Legendre collocation methods. Robin (mixed) boundary conditions on $\partial\Omega$ are enforced by a chosen set of control nodes distributed along $\partial\Omega$ in some fashion. The solution of the PDE at these control nodes satisfies the given boundary conditions forming a set of complementary constraint equations. Together with the solving operator, they form a global system of linear equations.