Analysis of a projection method for low-order nonconforming finite elements

We present a study of the incremental projection method to solve incompressible unsteady Stokes equations based on a low-degree nonconforming finite element approximation in space, with, in particular, a piecewise constant approximation for the pressure. The numerical method falls into the class of algebraic projection methods. We provide an error analysis in the case of Dirichlet boundary conditions, which confirms that the splitting error is second order in time. In addition, we show that pressure artificial boundary conditions are present in the discrete pressure elliptic operator, even if this operator is obtained by a splitting performed at the discrete level; however, these boundary conditions are imposed in the finite volume (weak) sense, and the optimal order of approximation in space is still achieved in numerical experiments, even for open boundary conditions.