  For incompressible flow models, the pressure term serves as a Lagrange
multiplier to ensure that the incompressibility constraint is satisfied. In
engineering applications, the pressure term is necessary for calculating
important quantities based on stresses like the lift and drag. For reduced
order models generated via a Proper orthogonal decomposition, it is common for
the pressure to drop out of the equations and produce a velocity-only reduced
order model. To recover the pressure, many techniques have been numerically
studied in the literature; however, these techniques have undergone little
rigorous analysis. In this work, we examine two of the most popular approaches:
pressure recovery through the Pressure Poisson equation and recovery via the
momentum equation through the use of a supremizer stabilized velocity basis. We
examine the challenges that each approach faces and prove stability and
convergence results for the supremizer stabilized approach. We also investigate
numerically the stability and convergence of the supremizer based approach, in
addition to its performance against the Pressure Poisson method.
