Spectral methods are a class of methods for solving PDEs numerically.
If the solution is analytic, it is known that these methods converge
exponentially quickly as a function of the number of terms used.
The basic spectral method only works in regular geometry (rectangles/disks).
A huge amount of effort has gone into extending it to
domains with a complicated geometry. Domain decomposition/spectral
element methods partition the domain into subdomains on which the PDE
can be solved (after transforming each subdomain into a
regular one). We take the dual approach - embedding the domain into
a larger regular domain - known as the fictitious domain method or
domain embedding. This method is extremely simple to implement and
the time complexity is almost the same as that for solving the PDE
on the larger regular domain. We demonstrate exponential convergence
for Dirichlet, Neumann and nonlinear problems. Time permitting, we
shall discuss extension of this technique to PDEs with discontinuous
coefficients.