  We propose a new least squares finite element method to solve the Stokes
problem with two sequential steps. The approximation spaces are constructed by
patch reconstruction with one unknown per element. For the first step, we
reconstruct an approximation space consisting of piecewise curl-free
polynomials with zero trace. By this space, we minimize a least
squaresfunctional toobtain thenumericalapproximationstothe gradientof
thevelocityand the pressure. In the second step, we minimize another least
squares functional to give the solution to the velocity in the reconstructed
piecewise divergence-free space. We derive error estimates for all unknowns
under L2 norms and energy norms. Numerical results in two dimensions and three
dimensions verify the convergence rates and demonstrate the great flexibility
of our method.
