  An essential ingredient of a spectral method is the choice of suitable bases
for test and trial spaces. On complex domains, these bases are harder to
devise, necessitating the use of domain partitioning techniques such as the
spectral element method. In this study, we introduce the Projection Extension
(PE) method, an approach that yields spectrally accurate solutions to various
problems on complex geometries without requiring domain decomposition. This
technique builds on the insights used by extension methodologies such as the
immersed boundary smooth extension and Smooth Forcing Extension (SFE) methods
that are designed to improve the order of accuracy of the immersed boundary
method. In particular, it couples an accurate extension procedure, that
functions on arbitrary domains regardless of connectedness or regularity, with
a least squares minimization of the boundary conditions. The resulting
procedure is stable under iterative application and straightforward to
generalize to higher dimensions. Moreover, it rapidly and robustly yields
exponentially convergent solutions to a number of challenging test problems,
including elliptic, parabolic, Newtonian fluid flow, and viscoelastic problems.
