Adaptivity techniques in High Performance FEM Simulation

During the last years, tremendous progress has been achieved in the development of methods for a posteriori error control and grid adaptation related to numerical methods for solving Partial Differential Equations (PDEs). It turned out that sophisticated adaptivity techniques are a key ingredient for fast and reliable numerical simulations. Hereby, the emphasis has been put on methods utilising selective refinement of grid cells (h-adaptivity) according to estimated error distributions. However, this leads to highly unstructured grids which decrease the numerical efficiency of an FEM code as these grids require many unaligned and costly memory accesses during the program run. Because of this, only a small fraction of the theoretically available CPU-power can be used in practical computations. Our FEM library FEAST [1,2] potentially overcomes this difficulty by using grids consisting of multiple logical tensor product grids. Logical tensor product grids are grids which are topologically equivalent to a tensor product mesh, i.e. every inner node has exactly four neighbouring mesh nodes (cf. Figure 1). These meshes unfortunately prevent applying standard h-adaptivity techniques. As a remedy, we apply patchwise h-adaptivity techniques and, more important, grid deformation methods (r-adaptivity) which relocate the grid points preserving the logical tensor product structure of the underlying mesh [3, 4]. In this article, we give an overview over the adaptivity and error control techniques applied in FEAST and compare them regarding the quality of the simulation. This is done by considering several prototypical test examples.