Is finite element anlysis accurate? How?

Most FEA software vendors provide verification samples that compare FEA results to classical closed-form solutions that are exact. Good FEA solvers deliver answers that compare to the exact answer within a couple of percentage points or better.

Engineering, the Art of Approximation and Inference

As engineers, we realize that in the real world, there are no exact answers, just very good approximations. Again, accuracy depends on what characterizes well executed FEA .