In the finite element method, we often construct the constraints of the system by adding penalty-function terms ( which often are many many magnitudes, up to $10^6$ order bigger than the largest stiffness matrix term) to the stiffness matrix by using the so-called Penalty Method (section 9.2)
On the other hand,
My hunch is that, the out-of-magnitude penalty term in stiffness matrix will make the whole Arnoldi iteration unstable, when it is iterated many times over, resulting in inaccuracy when we compute the eigenvalue analysis. I also don't know how to estimate the inaccuracy level .
Due to lack of competency in rigorous math, I am unable to prove my hunch. Am I right? Can anyone furnish a proof to prove or disprove my hunch?
PS: My hunch is somewhat validated by a note in the textbook reference above ( section 9.7),
But still, I'll be interested to see a rigorous mathematical argument.