Perfectly nested finite element spaces using generalized hanging variables

Summayr form only given. Starting from an initial mesh, discretizations of higher resolution can be constructed very easily by partitioning existing finite elements (FEs) recursively (Rude, U., 1993). Besides guaranteeing a lower bound on mesh quality, the resulting nested structure adds a high degree of regularity to the FE discretization, which greatly simplifies the application of fast geometrical multigrid solvers. At the same time, efficient error control requires the mesh size to adjust locally according to the spatial behavior of the electromagnetic fields and thus implies the need for strongly non-uniform mesh refinement. Our approach allows finite elements of unequal refinement levels to touch. As a result, the FE hierarchy is always perfectly nested. We focus on the computational efficiency of the generalized hanging variables framework. We first prove that, for the whole set of permissible constraints, the corresponding sequences of finite element spaces remain perfectly nested. We then propose one specific method which is easy to implement and which combines low memory requirements and rapid numerical convergence. Numerical examples are given to validate our findings.