Suppose I have a convex polygon $\Omega$ with $n$ sides and I want to find eigenvalues of the Laplacian with mixed Robin boundary conditions: $$ -\Delta u=k^2u\text{ in }\Omega $$ $$ a_iu+b_i\frac{\partial u}{\partial n}=0\text{ on }\Gamma_i $$
where $a_i,b_i$ are constant and $\partial\Omega=\cup_{i=1}^n\Gamma_i$. If I try and solve this via finite elements of equal smoothness and accuracy $p$, is it true that I gain order $2p$ for convergence to the eigenvalues and eigenfunctions?