Willoughby State Hospital is a psychiatric medical care facility based outside of Washington D.C. The hospital houses an acute care unit for troubled patients, run by Dr. Adam Fletcher, as well as waiting area with games and Internet access.
It is unclear whether Willoughby Hospital exists on Earth or not; it is almost certain the Asuran Niam fabricated Dr. Fletcher for his own purposes -- helping to convince Dr. Elizabeth Weir she required medication. In reality, Weir never left Atlantis, and the nanites in her body would spread when she gave in to her pills. (SGA: "The Real World")