The development of an finite element code which both offers sophisticated adaptivity techniques and features the possibility to efficiently exploit the computing power of the massively parallel supercomputers is a delicate task. The currently most widespread adaptive strategy, elementwise h-adaptivity, is known to lead to FEM-discretisations which are very efficient in terms of the number of unknowns. However, with current implementation techniques, only a small fraction of the peak performance can be exploited in practice. Moreover, using elementwise h-adaptivity implies the need for efficient strategies for load balancing. This is non-trivial as well.
Therefore, we propose another method of grid adaptation: grid deformation, i.e. the relocation of the grid points preserving the topological structure of the mesh. Following a sophisticated domain-decomposition multigrid approach (SCaRC), our finite element package FEAST works with grids consisting of many local generalised tensor product meshes. Their local regular structure leads to considerably faster computations compared with unstructured grids. During grid deformation, this local regular structure is preserved in contrast to elementwise h-adaptivity and thus our deformation method leads to ``performance-preserving`` adaptivity. Moreover, the issue of load-balancing is far less pronounced with grid deformation, as the number of unknowns per subdomain does not change during the deformation process.
Our newly developed grid deformation method needs the solve of a single Poisson problem and two fully decoupled initial value problems per grid point only. We will discuss mathematical analysis and numerical realisation of our method and investigate its asymptotic complexity and its accuracy with numerical tests. Moreover, we deal with its embedding into r- and rh-adaptive algorithms and their application to practical simulations.