For an above the knee amputation, many of the vessels can simply be ligated and left at that. The process the body uses is much simpler than neovascularization which seems to be getting upvoted in this thread for some reason.

The resulting pressure from a ligated dead end will increase vascular resistance greater than that of the systemic circulation. since blood is a fluid in a closed system, it loves to take the path of least resistance, and will simply bypass the ligated small arteries. Venous blood are capacitance vessels and can distend a large amount before reaching this point. Same principle applies however. After a while, you'll get thrombus formation in those vessels and you'll essentially have a plug where the blood isn't flowing.