Finite-element analysis - as far as I know - has the advantage of being able to estimate its own errors with some practically useful local upper bound. Because physics at that level is very much local. (Though it can't really help with higher level problems, like cyclic dynamical load and resonance problems, but there are other tools for that.)

> [...] most programming doesn't use static analysis tools beyond type systems.

Yes, unfortunately. :( It'd be great to have access to much better/stricter tools to be able to check just some interesting/important parts of programs (eg. pure business logic modules/functions) without resorting to reimplementing them in static-analysis systems.