A pressure-Poisson stabilized finite element method for the non-stationary Stokes equations to circumvent the inf-sup condition

In this article, stabilized finite element methods are considered for the non-stationary Stokes equations, based on some lowest equal-order finite elements space pair (Xh, Mh) which do not satisfy the discrete inf-sup condition. The stability of two kinds of methods is derived under some regularity assumptions. Then, the convergence of the penalty method and the pressure-Poisson stabilized method is compared. The result shows that the former error limits the order of approximation to O(ϵ+h/ϵ), and the latter yields the optimal error estimate O(h).