Adaptive anisotropic mesh and interface capturing problem

This talk will focus on some of the recent progresses in anisotropic mesh adaptation in finite element calculation. The main purpose of anisotropic meshing is to enable high stretched elements in the directional features of the searched solution. For particular simulation problem, we show a spectacular gain in accuracy while controlling the amount of calculation. We explain how anisotropic meshers are driven by a continuous metric field to measure length in a Riemannian way. Following the interpolation error estimation theory, the metric field must be related to the second derivative of representative field associated with the discrete solution. Afterwards, the anisotropic a posteriori error estimator drives the search of the optimal mesh (metric) that minimizes the error estimator under the constraint of a given number of nodes. We will show the necessary tools require achieving parallel anisotropic mesh adaptation: the metric construction and global objective function, the serial mesh generator (MTC) in a parallel context, the repartitioning the mesh after each re-meshing stage. Numerical 2D and 3D applications are presented here to show that the proposed anisotropic error estimator gives an accurate representation of the exact error. We will show also, that the optimal adaptive mesh procedure provides a mesh refinement and element stretching which appropriately captures interfaces for practical application problems that are unreachable by simply refining the meshes