  We propose and analyze a pressure-stabilized projection Lagrange--Galerkin
scheme for the transient Oseen problem. The proposed scheme inherits the
following advantages from the projection Lagrange--Galerkin scheme. The first
advantage is computational efficiency. The scheme decouples the computation of
each component of the velocity and pressure. The other advantage is essential
unconditional stability. Here we also use the equal-order approximation for the
velocity and pressure, and add a symmetric pressure stabilization term. This
enriched pressure space enables us to obtain accurate solutions for small
viscosity. First, we show an error estimate for the velocity for small
viscosity. Then we show convergence results for the pressure. Numerical
examples of a test problem show higher accuracy of the proposed scheme for
small viscosity.
