Anisotropic mesh adaption based on a posteriori estimates and optimisation of node positions

Boundary or interior layers are usually highly directional solution features. Thus, suitable anisotropic meshes, reflecting the directional features of the solution, provide the basis for the most efficient numerical approximation. Anisotropic mesh design strategies based upon a priori analysis have been developed for a variety of PDE problems and discretisations. On the other hand a posteriori error estimation techniques have been developed and integrated with mesh refinement strategies, leading to numerical methods which perform extremely well over broad classes of problems, even when no a priori analysis is available. One particular advantage of the a posteriori approach is that it can yield meshes that efficiently approximate specific functionals of the solution [2]. Most of the common a posteriori based algorithms are unable to introduce suitable anisotropy into the mesh however. We introduce here a new mesh adaptation strategy which allows suitable anisotropy within the mesh. The approach draws upon methods from numerical optimisation in order to modify the node positions of a given (isotropic) mesh such that an a posteriori error estimate is minimised. To make this feasible for non-trivial problems the discrete adjoint technique [3] is utilised to efficiently evaluate the gradient of the a posteriori error estimate. The Dual Weighed Residual (DWR) error estimate [2] for the error in a quantity of interest is utilised to allow goal driven adaptivity. This present paper is based upon our previous work [5] to which refer for a more detailed consideration of the approach and for an overview of related previous work.