  We consider the effectiveness of foreground cleaning in the recovery of
Cosmic Microwave Background (CMB) polarization sourced by gravitational waves
for tensor-to-scalar ratios in the range $0<r<0.1$. Using the planned survey
area, frequency bands, and sensitivity of the Cosmology Large Angular Scale
Surveyor (CLASS), we simulate maps of Stokes $Q$ and $U$ parameters at 40, 90,
150, and 220 GHz, including realistic models of the CMB, diffuse Galactic
thermal dust and synchrotron foregrounds, and Gaussian white noise. We use
linear combinations (LCs) of the simulated multifrequency data to obtain
maximum likelihood estimates of $r$, the relative scalar amplitude $s$, and LC
coefficients. We find that for 10,000 simulations of a CLASS-like experiment
using only measurements of the reionization peak ($\ell\leq23$), there is a 95%
C.L. upper limit of $r<0.017$ in the case of no primordial gravitational waves.
For simulations with $r=0.01$, we recover at 68% C.L.
$r=0.012^{+0.011}_{-0.006}$. The reionization peak corresponds to a fraction of
the multipole moments probed by CLASS, and simulations including
$30\leq\ell\leq100$ further improve our upper limits to $r<0.008$ at 95% C.L.
($r=0.01^{+0.004}_{-0.004}$ for primordial gravitational waves with $r=0.01$).
In addition to decreasing the current upper bound on $r$ by an order of
magnitude, these foreground-cleaned low multipole data will achieve a cosmic
variance limited measurement of the E-mode polarization's reionization peak.
