  The recent development of spectral method has been praised for its high-order
convergence in simulating complex physical problems. The combination of
embedded boundary method and spectral method becomes a mainstream way to tackle
geometrically complicated problems. However, the convergence is deteriorated
when embedded boundary strategies are employed. Owing to the loss of
regularity, in this paper we propose a new spectral collocation method which
retains the regularity of solutions to solve differential equations in the case
of complex geometries. The idea is rooted in the basis functions defined in an
extended domain, which leads to a useful upper bound of the Lebesgue constant
with respect to the Fourier best approximation. In particular, how the
stretching of the domain defining basis functions affects the convergence rate
directly is detailed. Error estimates chosen in our proposed method show that
the exponential decay convergence for problems with analytical solutions can be
retained. Moreover, two-dimensional Poisson equations and convection-diffusion
equations with simple and complex geometrical domains will be simulated. The
predicted results justify the advantages of applying our method to tackle
geometrically complicated problems.
