In the TV series Under the Dome, the military used a MOAB (Mother of all Bombs) to remove the dome problem. In episode 2 one old man tried using digging tractor but Big Jim stopped him because it could explode. Why didn't they try digging into the ground? The military could use non-electronic methods, like lots of men with shovels. Is there any explanation of it in the novel?
At the beginning of the second episode (about 3 minutes in), several people are seen digging with spades, near Joe's house:
Followed by this dialogue:
Joe: Any luck yet?
Man: No, this thing goes down far.
As GµårÐïåñ's answer states, in the third episode Junior (being followed by Julia) go into the tunnels and find the dome goes really far down.
As was revealed in the scene with Junior Rennie going into the tunnels in an attempt to find a way out by going under the dome, the dome extends down into the ground. For all intents and purposes, the assumption is that the dome extends down into the ground far enough that one cannot dig or if they do, it will extend into that void as well. However, I have to admit that seeing the smaller dome in the woods dug up and relocated to the barn, it would stand to reason the same should be doable with the larger dome, although probably to a much greater scale which might be out of the reach of the people inside.