Adaptive time-step with anisotropic meshing for incompressible flows

This talk focuses on the development of a new space and time adaptive method for incompressible flows. We start with an overview about the anisotropic mesh adaptation, introduced in [1]. The latter method is developed using a posteriori estimates relying on the length distribution tensor approach and the associated edge based error analysis. The key point of this work is the extension of the mesh adaptation technique to contain adaptive time advancing based on a newly developed time error estimator [2]. The objective is to show that the combination of time and space anisotropic adaptations with highly stretched elements can be used to compute high Reynolds number flows within reasonable computational and storage costs. In particular, it will be shown in the numerical experiments that boundary layers, flow detachments and all vortices are well captured automatically by the mesh. The time-step is controlled by the interpolation error and preserves the accuracy of the mesh adapted solution. A Variational MultiScale (VMS) method [3] is employed for the discretization of the Navier-Stokes equations. Numerical solutions of some benchmark problems demonstrate the applicability of the proposed space-time error estimator. An important feature of the proposed method is its conceptual and computational simplicity as it only requires from the user a number of nodes according to which the mesh and the time-steps are automatically adapted.