A posteriori error estimation and anisotropy detection with the dual weighted residual method

In this work we develop a new framework for a posteriori error estimation and detection of anisotropies based on the dual weighted residual method by Becker and Rannacher. The common approach for anisotropic mesh adaptation is to analyze the Hessian of the solution. Eigenvalues and eigenvectors indicate dominant directions and optimal stretching of elements. However this approach is firmly linked to energy norm error estimation. Here, we extend the dual weighted residual method to anisotropic finite elements allowing for the direct estimation of directional errors with regard to given output functionals. The resulting meshes reflect anisotropic properties of both the solution and the functional. For the optimal measurement of the directional errors, the coarse meshes need some alignement with the dominant anisotropies. Numerical examples will demonstrate the efficiency of this method on various three dimensional problems including a well known Navier-Stokes benchmark.