$k\ell$-refinement: An adaptive mesh refinement scheme for hiearchical hybrid grids

Abstract: This work introduces an adaptive mesh refinement technique for hierarchical hybrid grids with the goal to reach scalability and maintain excellent performance on massively parallel computer systems. On the block structured hierarchical hybrid grids, this is accomplished by using classical, unstructured refinement only on the coarsest level of the hierarchy, while keeping the number of structured refinement levels constant on the whole domain. This leads to a compromise where the excellent performance characteristics of hierarchical hybrid grids can be maintained at the price that the flexibility of generating locally refined meshes is constrained. Furthermore, mesh adaptivity often relies on a posteriori error estimators or error indicators that tend to become computationally expensive. Again with the goal of preserving scalability and performance, a method is proposed that leverages the grid hierarchy and the full multigrid scheme that generates a natural sequence of approximations on the nested hierarchy of grids. This permits to compute a cheap error estimator that is well-suited for large-scale parallel computing. We present the theoretical foundations for both global and local error estimates and present a rigorous analysis of their effectivity. The proposed method, including error estimator and the adaptive coarse grid refinement, is implemented in the finite element framework HyTeG. Extensive numerical experiments are conducted to validate the effectiveness, as well as performance and scalability.