They are basically a stronger version of Trooper Veteran.
They should not be confused with the "Trooper Elites" in Valkyria Chronicles and Valkyria Chronicles 4, who are just levelled-up members of the regular Shocktrooper Class rather than a specific sub-class. Shocktroopers simply gain the label "Elite" when they reach level 11 in these games.
Gallian Trooper Elites
Valkyria Chronicles 2
Valkyria Chronicles 3
When a shocktrooper classes up to a trooper elite, they are able to equip flamethrowers, replacing the role of commando from Valkyria Chronicles 2.
GRA Trooper Elites
The GRA have no known Trooper Elites.