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An element based adaptation method is developed for an anisotropic aposteriori error estimator. The adaptation does not make use of a metric, butinstead equidistributes the error over elements using local mesh modifications.Numerical results are reported, comparing with three popular anisotropicadaptation methods currently in use. It was found that the new method givesfavourable results for controlling the energy norm of the error in terms ofdegrees of freedom at the cost of increased CPU usage. Additionally, weconsidered a new $L^2$ variant of the estimator. The estimator is shown to beconditionally equivalent to the exact $L^2$ error. We provide examples ofadapted meshes with the $L^2$ estimator, and show that it gives greater controlof the $L^2$ error compared with the original estimator.

Anisotropic residual based a posteriori mesh adaptation in 2D: element based approach