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

Abstract

We consider a family of tensor product finite element methods for hyperbolic equations in RN, N ≥ 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 ≥ 3, the corresponding methods are unstable.