  Standard ROMs generally yield spurious numerical oscillations in the
simulation of convection-dominated flows. Regularized ROMs use explicit ROM
spatial filtering to decrease these spurious numerical oscillations. The Leray
ROM is a recently introduced regularized ROM that utilizes explicit ROM spatial
filtering of the convective term in the Navier-Stokes equations.
  This paper presents the numerical analysis of the finite element
discretization of the Leray ROM. Error estimates for the ROM differential
filter, which is the explicit ROM spatial filter used in the Leray ROM, are
proved. These ROM filtering error estimates are then used to prove error
estimates for the Leray ROM. Finally, both the ROM filtering error estimates
and the Leray ROM error estimates are numerically investigated in the
simulation of the two-dimensional Navier-Stokes equations with an analytic
solution.
