Rene Schneider (TU Chemnitz)

Toward anisotropic mesh adaptation based upon sensitivity of a posteriori estimates
Many problems to which adaptive mesh refinement is applied possess anisotropic solution features. These features are best approximated on an anisotropic mesh, yet the adaptive refinement algorithms applied today do only support locally isotropic refinement, which leads to overrefinement. Automatic anisotropic mesh refinement is still in its infancy, with most approaches lacking generality or efficiency.

We present a new approach to this problem basing on optimisation of an a posteriori error estimate with respect to the node positions in a finite element mesh. This approach becomes feasible because the derivatives of such an error estimate with respect to the node positions can be evaluated efficiently by means of the discrete adjoint technique. Availability of these derivatives enables the use of efficient algorithms from mathematical optimisation.

The basic approach is discussed and applications to singularly perturbed reaction-diffusion and convection-diffusion problems are demonstrated.