  Simulation of unsteady creeping flows in complex geometries has traditionally
required the use of a time-stepping procedure, which is typically costly and
unscalable. To reduce the cost and allow for computations at much larger
scales, we propose an alternative approach that is formulated based on the
unsteady Stokes equation expressed in the time-spectral domain. This
transformation results in a boundary value problem with an imaginary source
term proportional to the computed mode that is discretized and solved in a
complex-valued finite element solver using Bubnov-Galerkin formulation. This
transformed spatio-spectral formulation presents several advantages over the
traditional spatio-temporal techniques. Firstly, for cases with boundary
conditions varying smoothly in time, it provides a significant saving in
computational cost as it can resolve time-variation of the solution using a few
modes rather than thousands of time steps. Secondly, in contrast to the
traditional time integration scheme with a finite order of accuracy, this
method exhibits a super convergence behavior versus the number of computed
modes. Thirdly, in contrast to the stabilized finite element methods for fluid,
no stabilization term is employed in our formulation, producing a solution that
is consistent and more accurate. Fourthly, the proposed approach is
embarrassingly parallelizable owing to the independence of the solution modes,
thus enabling scalable calculations at a much larger number of processors. The
comparison of the proposed technique against a standard stabilized finite
element solver is performed using two- and three-dimensional canonical and
complex geometries. The results show that the proposed method can produce more
accurate results at 1% to 11% of the cost of the standard technique for the
studied cases.
