  An element based adaptation method is developed for an anisotropic a
posteriori error estimator. The adaptation does not make use of a metric, but
instead equidistributes the error over elements using local mesh modifications.
Numerical results are reported, comparing with three popular anisotropic
adaptation methods currently in use. It was found that the new method gives
favourable results for controlling the energy norm of the error in terms of
degrees of freedom at the cost of increased CPU usage. Additionally, we
considered a new $L^2$ variant of the estimator. The estimator is shown to be
conditionally equivalent to the exact $L^2$ error. We provide examples of
adapted meshes with the $L^2$ estimator, and show that it gives greater control
of the $L^2$ error compared with the original estimator.
