Black Eleven was a shipyard that belonged to the Imperial Navy under the command of Black Sword Command, but later the Yevetha took control of it.
During the Imperial occupation, Davith Sconn visited the Koornacht Cluster, and Black Eleven was situated on the Yevethan world of Zhina. In 4 ABY, which was eight months following the Battle of Endor, the Galactic Empire withdrew from the Duskhan League systems. The Yevetha then seized the opportunity to capture several Imperial ships and installations, including Black Eleven, which was later moved from Zhina to a more secure location.
By 16 ABY, the Yevetha relocated the Black Eleven Fleet to the Doornik-1142 system, which is located on the edge of the Koornacht Cluster. The New Republic scout ship Astrolabe almost discovered this hidden base when it entered the system for a mapping mission at the beginning of the Black Fleet Crisis. However, the battle cruiser Purity destroyed the Astrolabe to protect the fleet. Davith Sconn informed the New Republic about Black Eleven, but initial military reconnaissance missions by the 21st Recon Wing after the Battle of Doornik-319 failed to locate it.
However, the Purity's actions ultimately revealed the Doornik-1142 base to the New Republic. Admiral Ackbar deduced that the Astrolabe had stumbled upon one of the former Imperial shipyards. Consequently, in 17 ABY, the New Republic Fifth Fleet dispatched a squadron of capital ships to Doornik-1142 to investigate.
The Shield of Lies novel, the second installment in the Black Fleet Crisis trilogy, briefly mentions the Black Eleven shipyard. It states that the facility was orbiting Zhina during the Imperial occupation. The Black Fleet novels indicate that the Yevetha captured several Imperial shipyards in the Koornacht Cluster, but it's not definitively stated whether Black Eleven was among them.
In the same passage where Black Eleven is mentioned, Admiral Ackbar concludes that the destruction of the Astrolabe at Doornik-1142 (an event from the previous novel, Before the Storm) revealed the presence of a captured shipyard. In Tyrant's Test, the third novel in the series, there's a brief mention of the New Republic sending a strong force of capital ships to investigate this possibility.
The inclusion of Black Eleven among the shipyards captured by the Yevetha is explicitly stated in several secondary reference works, starting with the 1998 Star Wars Encyclopedia and continuing in later works such as the 2008 The Complete Star Wars Encyclopedia.
These reference sources also identify several Imperial warships captured by the Yevetha as part of a "Black Eleven Fleet" based at Doornik-1142. This includes the Purity of Yevetha, which served as a guard ship in that system in Before the Storm, as well as the Victory-class Star Destroyer Valorous, and the Dreadnaught and corvette renamed Blessings and Beauty of Yevetha.
It is uncertain whether these details came from behind-the-scenes notes from Michael P. Kube-McDowell, the author of the Black Fleet novels, or whether Stephen J. Sansweet or another contributor to the Encyclopedia made these inferences. Some related continuity references to the Black Sword ships in these sources appear to be incorrect, such as the claim that the Beauty of Yevetha was destroyed during the Battle of Doornik-319.
While the presence of the "Black Eleven Fleet" at Doornik-1142 suggests that the shipyard was also located there, the Encyclopedia doesn't explicitly state this, and Tyrant's Test doesn't mention whether the reconnaissance mission discovered the shipyard. Instead, it focuses on the discovery of Black Nine in the ILC-905 system by elements of Task Force Aster, and the subsequent battle there.