  A modified Chorin-Teman (Euler non-incremental) projection method and a
modified Euler incremental projection method for non inf-sup stable mixed
finite elements are analyzed. The analysis of the classical Euler
non-incremental and Euler incremental methods are obtained as a particular
case. We first prove that the modified Euler non-incremental scheme has an
inherent stabilization that allows the use of non inf-sup stable mixed finite
elements without any kind of extra added stabilization. We show that it is also
true in the case of the classical Chorin-Temam method. For the second scheme,
we study a stabilization that allows the use of equal-order pairs of finite
elements. The relation of the methods with the so called pressure stabilized
Petrov Galerkin method (PSPG) is established. The influence of the chosen
initial approximations in the computed approximations to the pressure is
analyzed. Numerical tests confirm the theoretical results.
