On the stability of a family of finite element methods for hyperbolic problems

We consider a family of tensor product finite element methods for hyperbolic equations in $R^{N}$, $N\ge 2$, which are explicit and generate a continuous approximate solution. The base case $N=2$ (an extension of the box scheme to higher order) is due to Winther, who proved stability and optimal order convergence. By means of a simple counterexample, we show that, for linear approximation with $N \ge 3$, the corresponding methods are unstable.