  We show that it is possible to obtain a linear computational cost FEM-based
solver for non-stationary Stokes and Navier-Stokes equations. Our method
employs a technique developed by Guermond and Minev, which consists of singular
perturbation plus a splitting scheme. While the time-integration schemes are
implicit, we use finite elements to discretize the spatial counterparts. At
each time-step, we solve a PDE having weak-derivatives in one direction only
(which allows for the linear computational cost), at the expense of handling
strong second-order derivatives of the previous time step solution, on the
right-hand side of these PDEs. This motivates the use of smooth functions such
as B-splines. For high Reynolds numbers, some of these PDEs become unstable. To
deal robustly with these instabilities, we propose to use a residual
minimization technique. We test our method on problems having manufactured
solutions, as well as on the cavity flow problem.
