Unsymmetric multi-level hanging nodes and anisotropic polynomial degrees in H1-conforming higher-order finite element methods

The implementation of higher-order finite element schemes that can handle multi-level hanging nodes is known to be a difficult task. In fact, most of the available literature on hanging nodes in finite element schemes restricts to one-level hanging nodes resulting from symmetric bisections. The intent of this paper is to provide all data structures and algorithms that are necessary for an implementation of a H1-conforming higher-order finite element method. The corresponding finite element spaces are defined via tensor products of hierarchic as well as nodal polynomials for quadrilateral and hexahedron based meshes with unsymmetric, multi-level hanging nodes and arbitrary anisotropic polynomial degree distributions, where special care is given to possible orientation problems. The meshes may even result from non-matching refinements. Given these data structures and algorithms, an extension for Serendipity spaces is described in detail along with some other techniques to improve computational efficiency. Numerical results from an implementation based on these data structures and algorithms serve as a validation and show the broad possibilities that highly flexible higher-order finite element schemes have to offer. To the best of our knowledge, this paper offers the most comprehensive numerical results so far for various three-dimensional benchmark problems using in particular finite element spaces defined for hierarchical as well as nodal polynomials on meshes with unsymmetric refinement ratios as well as (multi-level) hanging nodes. Most notably, the numerical results indicate that unsymmetric refinements are indeed favorable over symmetric refinements with respect to convergence rates. However, the actual optimal refinement ratio for a given problem seems to depend on the type and magnitude of singularities to be resolved as well as on the chosen (full) tensor product or Serendipity finite element spaces. In addition to these numerical results, we find that systems of equations defined via finite element spaces using the nodal Lagrange polynomials with Gauss-Lobatto quadrature points as support points yield drastically improved condition numbers.