A naturally parallelizable formulation is considered for solving linearized time-dependent Navier-Stokes equations. The evolution problem is first converted into a complex valued elliptic system by Fourier transformation. Existence and uniqueness are then given for the resulting problem for each frequency. Stability and regularity depending on frequency are analyzed. Next, standard finite element methods are used to approximate solutions for the transformed elliptic systems. Finally, time-dependent solutions are constructed by Fourier inversion with a full estimate of errors generated in the truncation in the Fourier transformation, quadrature rules, and finite element approximations. (C) 2000 Published by Elsevier Science S.A. All rights reserved.