Hierarchical high order finite element bases for H(div) spaces based on curved meshes for two-dimensional regions or manifolds

• Innovative two dimensional Hdiv approximation spaces for curved quadrilateral and triangular elements.
• Hdiv bases properly constructed on the master element and mapped to the geometrical elements by the Piola transformation, followed by a normalization procedure.
• Stable convergence with optimal convergence rates, coinciding for primal and dual variables.
• Study of number of condensed equations as a function of the polynomial order of the shape functions.

The mixed finite element formulation for elliptic problems is characterized by simultaneous calculations of the potential (primal variable) and of the flux field (dual variable). This work focuses on new H(div)-conforming finite element spaces, which are suitable for flux approximations, based on curved meshes of a planar region or a manifold domain embedded in R3R3. The adopted methodology for the construction of H(div) bases consists in using hierarchical H1H1-conforming scalar bases multiplied by vector fields that are properly constructed on the master element and mapped to the geometrical elements by the Piola transformation, followed by a normalization procedure. They are classified as being of edge or internal type. The normal component of an edge function coincides on the corresponding edge with the associated scalar shape function, and vanishes over the other edges, and the normal components of an internal shape function vanishes on all element edges. These properties are fundamental for the global assembly of H(div)-conforming functions locally defined by these vectorial shape functions. For applications to the mixed formulation, the configuration of the approximation spaces is such that the divergence of the dual space and the primal approximation space coincides. Results of verification numerical tests are presented for curved triangular and quadrilateral partitions on circular, cylindrical and spherical regions, demonstrating stable convergence with optimal convergence rates, coinciding for primal and dual variables.