  Discretizing a solution in the Fourier domain rather than the time domain
presents a significant advantage in solving transport problems that vary
smoothly and periodically in time, such as cardiorespiratory flows. The finite
element solution of the resulting time-spectral formulation is investigated
here for the convection-diffusion equations. In addition to the baseline
Galerkin's method, we consider stabilized approaches inspired by the streamline
upwind Petrov/Galerkin (SUPG), Galerkin/least square (GLS), and variational
multiscale (VMS) methods. We also introduce a new augmented SUPG (ASU) method
that, by design, produces a nodally exact solution in one dimension for
piecewise linear interpolation functions. Comparing these five methods using
1D, 2D, and 3D canonical test cases shows while the ASU is most accurate
overall, it exhibits stability issues in extremely oscillatory flows with a
high Womersley number in 3D. The GLS method, which is identical to the VMS for
this problem, presents an attractive alternative due to its excellent stability
and reasonable accuracy.
