ogelius finite elements). For the Scott--Vogelius finite elements, we 
find the approximate eigenvalues are extremely sensitive if the mesh has nearly singular vertices, i.e.,
vertices that fall on exactly two ``almost'' straight lines (cf.~Section \ref{sec-SV}).  We give numerical examples that illustrate this behavior.

\revj{The analysis of composite triangulations (e.g. Clough-Tocher and Powell-Sabin) on the problem \eqref{weak} goes back at least to the work of Costabel and Dauge \cite{CoDa02}. Recently,  Duan et al. \cite{Duan,Duan19B,Duan20} considered Lagrange finite elements for Maxwell's eigenvalue problem in two and three dimensions using composite triangulations.  However, as noted earlier, they use a different formulation than the standard one \eqref{fem}.  In particular, in \cite{Duan20}  they add a Lagrange multiplier and an equation of the form appears $({\ensuremath\mathop{\mathrm{div}\,}} \bld{u}_h, q_h)=0$ which can be thought of as a Kikuchi-type formulation \cite{Kikuchi}, where one transfers the derivatives to $\bld{u}_h$. In \cite{Duan19B} a similar formulation is used with a regularization term.

As mentioned above, the main idea
to show convergence of Lagrange elements using the standard formulation \eqref{weak} on certain triangulations is 
the construction of a Fortin-type operation with certain approximation properties.
On certain composite triangulations (e.g. Powell-Sabin, Clough-Tocher, Alfeld, Worsey-Farin) exact sequences and/or Fortin projections have been developed; see for example \cite{GLN19, GLN20, FGN20, Zhang3d, QinThesis, ChristiansenHu}. These results have led to stable finite element for fluid flow problems; see for example \cite{Neilan}. In this paper, for the Powell-Sabin and Clough-Tocher triangulations, we cannot directly use  the Fortin projections defined in \cite{GLN19, FGN20} since they require too much smoothness. Instead, we pre-process with a Scott-Zhang type interpolant that preserves the vanishing tangential components, and then use the degrees of freedom  in \cite{GLN19, FGN20}. These  projections are sufficient for our purposes, however, it would be very interesting to see if one can construct $L^2$ bounded commuting projection for these sequences as is done in the finite element exterior calculus (FEEC) \cite{Christiansen}. 
If bounded $L^2$ bounded commuting projections exist, then
the convergence of eigenvalue problems  follows from the general theory in FEEC \cite{AFW2006, ArnoldEtal10, BoffiActa}. 
}

 


  


 
The paper is organized as follows: In the next section we give  a convergence
proof for finite elements spaces with stable projections. In \DB
Section~\ref{sec-Examples},\BD{}
we provide three examples of Lagrange finite element spaces with stable projections:  the piecewise linear Lagrange space
on Powell--Sabin splits, the piecewise quadratic Lagrange space on Clough--Tocher splits,
and the piecewise $k$th degree Lagrange space on generic triangulations.
\rev{In Section \ref{sec-Numerics} we provide numerical experiments
and make some concluding remarks in Section \ref{sec-conclude}.}



\section{Convergence Framework}
Define the two-dimensional ${\bld{\curl}}$, ${\rm rot}$, and divergence operators as
\[
{\bld{\curl}} u = \big(\frac{{\partial} u}{{\partial} x_2}, -\frac{{\partial} u}{\rev{{\partial} x_1}}\big)^\intercal,\qquad
{\rm rot}\, \bld{v}  = \frac{{\partial} v_2}{{\partial} x_1} - \frac{{\partial} v_1}{{\partial} x_2},\qquad \MJN{{\rm div}\, \bld{v} = \frac{{\partial} v_1}{{\partial} x_1} + \frac{{\partial} v_2}{{\partial} x_2}},
\]
\MJN{and define the Hilbert spaces
\begin{align*}
\bld{H}_0({\rm rot},\Omega) &= \{\bld{v}\in \bld{L}^2(\Omega):\ {\rm rot}\, \bld{v}\in L^2(\Omega),\ \bld{v}\cdot \bld{t}|_{{\partial}\Omega} = 0\},\\
\bld{H}({\rm div},\Omega) & = \{\bld{v}\in \bld{L}^2(\Omega):\ {\rm div}\, \bld{v}\in L^2(\Omega)\},
\end{align*}
where $\bld{t}$ is a unit tangent vector of ${\partial} \Omega$.
Recall that $\mathring{\bV} = \bld{H}_0({\rm rot},\Omega)\cap \bld{H}({\rm div},\Omega)$.
}

Let $\mathring{\bV}_h\subset \bld{H}_0({\rm rot},\Omega)$ and $\mathring{Q}_h \subset L_0^2(\Omega)$ be finite element spaces such that $ {\rm rot}\, \mathring{\bV}_h \subset \mathring{Q}_h$.  




\subsection{Source problems}
We will require the corresponding source problems for the analysis. 
\MJN{To this end, we define the solution operators ${\bm A}:L^2(\Omega)\to \bld{H}_0({\rm rot},\Omega)$
and $T:L^2(\Omega)\to L^2_0(\Omega)$ such that for given $f\in L^2(\Omega)$, there holds}
\begin{subequations}\label{source}
\begin{alignat}{2}
({\bm A} f, {\bm \tau})+(T f, {\rm rot}\, {\bm \tau})&= 0, \quad && \forall {\bm \tau} \in \bld{H}_0({\rm rot}, \Omega), \\
({\rm rot}\, {\bm A} f, q)&=(f,q)  \quad && \forall q \in L_0^2(\Omega).
\end{alignat}
\end{subequations}

Likewise, the discrete source problem is given by:
\begin{subequations}\label{femsource}
Find ${\bm A}_h f \in \mathring{\bV}_h$ and $T_h f \in \mathring{Q}_h$ such that
\begin{alignat}{2}
({\bm A}_h f, {\bm \tau}_h)+(T_h f, {\rm rot}\, {\bm \tau}_h)&= 0 \quad && \forall {\bm \tau} \in  \mathring{\bV}_h,  \\
({\rm rot}\, {\bm A}_h f, q_h)&=(f,q_h) \quad &&  \forall q_h \in \mathring{Q}_h.
\end{alignat}
\end{subequations}
Note that ${\bm A} f={\bld{\curl}} Tf$, and so ${\ensuremath\mathop{\mathrm{div}\,}} {\bm A} f=0$. Moreover, using that ${\rm rot}\, {\bm A} f=f$ we have that $\rev{{\bm A}}f \in \mathring{\bV}$. 

We define the operator norm:
\begin{equation}
\|T-T_h\|:=\sup_{f \in L^2(\Omega)\backslash \{0\}} \frac{\|(T-T_h) f\|_{L^2(\Omega)}}{\|f\|_{L^2(\Omega)}}.
\end{equation}
We will use the next standard result that states that the uniform convergence of the
discrete source problem implies convergence of the discrete eigenvalues.
\DB
This result is a consequence of the classical discussion
in~\cite[Section~8]{BaOs} (see also~\cite[Section~9]{BoffiActa}
and~\cite[Theorem~4.4]{bfgp}).
\BD
\begin{proposition}\label{uniform}
Let $T$ and $T_h$ be defined from \eqref{source} and \eqref{femsource}, respectively, and
  suppose that $\|T-T_h\| \rightarrow 0$ as $h \rightarrow 0$. Consider the problem \eqref{mixedweak}  
  and consider the nonzero eigenvalues $0 < \lambda^{(1)} \le \lambda^{(2)} \le \cdots$. Consider also  \eqref{mixedfem} and its non-zero eigenvalues  $0< \lambda_h^{(1)} \le \lambda_h^{(2)} \le \cdots$. Then, for any fixed $i$, $\lim_{h \rightarrow 0} \lambda_h^{(i)} =\lambda^{(