Is there any method for making the initial stages of an abseil smoother when the anchor in use is at foot level and the ledge drops off abruptly. I have had a big swing before when trying to do this and found it quite unpleasant.
I was always using and teaching the method, which Ben Crowell described very well in another answer, and it was standard in our club for a long time. However, over time it showed some important disadvantages.
First, it doesn't work reliably on slippery surface. When you "sit down", the force applied to your feet is high, and if one of them slips, you lose balance completely and usually hit the wall. And this hit may land on your controlling hand, so only the belay device (Prusik or something else which works without your help) will save you. I have had such and accident once and seen a lot more. The experience is quite nasty especially given that Prusik doesn't catch instantly.
Second, it's quite hard to finish your "sit down" movement smoothly, usually there is a small fall in the end. At this moment, the rope hits the sharp corner with quite a force, so it can be damaged. Luckily, I haven't seen the rope completely break off, but I've seen the sheath damaged as a result of such a move.
And if you are in the real mountains using real anchors, they may fall off, being subject to a bump.
So what should you do?
NB: you controlling hand should always be on your rope during the operation, regardless of what is happening.
If you are not sure, fix you rappelling device at a point, where you think you will be already hanging. By fixing I mean fixing in some way which works when you completely drop your hands. If you doesn't know how to do this with your device, please get to know it. As a last resort, you can just tie a butterfly knot and clip to it with a carabiner.
When training all this, use additional belay rope, controlled by someone else.
Some rappels are simply awkward to start. The awkwardness isn't the most important issue in and of itself, but you want to avoid a situation in which the start is so messy that you get banged around, it causes you to lose your grip on your brake hand, and you fall to your death. To prevent this, you can use a Prusik backup below the rappelling device; Freedom of the Hills has a description of how to do this on p. 198. (Other methods include a fireman's belay if you're not the first off, or being lowered if you're not the last off.)
Anyway, in terms of the sheer awkwardness of the start, you can try to compensate by putting your feet far apart and bending your knees as much as possible.
When I use the Prusik backup, I extend my rappelling device (ATC) so that the Prusik can't get caught in the ATC. This means that I normally have both hands below the ATC, on the brake strand. However, you can always take the left hand off of the brake strand if you think the use of that hand would help you to get over the edge.
Ryley raises a good point in a comment, which is that the extension actually forces you to get lower on the ledge before you can weight the rope. This is one reason that the system I prefer is slightly different from the one shown in figure 11-13 on p. 198 of FotH (8th ed). Rather than extending my ATC by a large amount (maybe 25 cm in their figure), I only extend it by about 10 cm, and I clip the Prusik into a leg loop. Because the Prusik is on the brake strand, it only needs to supply a small amount of force, so IMO attaching it to the leg loop is no less safe than attaching it to the belay loop. Using the leg loop allows the extension to be much shorter.