# Spurious eigenvalues in a finite element eigenvalue problem

This post is closely related to this one and uses the exact same setup: a mix of quadratic and cubic basis functions in a finite element approach, where variables $$u_1$$ and $$u_2$$ are quadratic and $$u_3$$ is cubic. Dirichlet boundary conditions apply for both $$u_2$$ and $$u_3$$ on both sides of the domain, where they equal zero.

In some cases I come across spurious eigenvalues (e.g. as described here) in solving the eigenvalue problem. Turns out I can actually remove them by setting both rows/columns of $$u_2$$ and $$u_3$$ to zero on the left edge instead of just the odd/even ones (again, see this post). Then the spectrum has no spurious eigenvalues, and the eigenfunctions still look correct. However, doing the same thing on the right edge messes up the eigenfunctions, showing a strange oscillation on the right side and the eigenfunctions no longer go to zero there. I also found out that doing it like this does not always fix the issue, for some setups the spurious eigenvalues still pop up.

## The question

I can't figure out why modifying both rows/columns on the left edge seems to remove the spurious eigenvalues in some cases, that is, are the quadratic/cubic basis functions related to this somehow? On the other hand, it may well be an unavoidable problem for particular setups, so in that case is there a way to filter them out or check the validity of the eigenvalues? I can't just remove the outer points, since their appearance is problem dependent and I don't know beforehand if they will show up or not.