We undertake the numerical analysis of a reaction-diffusion system of ‘$\lambda$–$\omega$’ type [26]. Results are presented for a fully-practical piecewise linear finite element method by mimicking results in the continuous case [11]. We establish a priori estimates and error bounds for a semi-discrete and a fully discrete finite element approximation. The theoretical results are illustrated and verified via the numerical solution of periodic plane waves in one space dimension. Experiments in two space dimensions led to‘target patterns’ and spiral wave break-up.