A lowest-order composite finite element exact sequence on pyramids
This work provides a missing piece for conforming finite element discretizations on meshes containing pyramids, enabling seamless coupling of hexahedral and tetrahedral elements in computational electromagnetics and related fields.
The authors construct lowest-order composite pyramidal finite elements that form an exact sequence satisfying the de Rham diagram and commuting property, and are fully compatible with standard Raviart-Thomas-Nédélec elements on tetrahedra and hexahedra. They show that these elements achieve optimal-order approximation in $h$ on mixed partitions.
Composite basis functions for pyramidal elements on the spaces $H^1(Ω)$, $H(\mathrm{curl},Ω)$, $H(\mathrm{div},Ω)$ and $L^2(Ω)$ are presented. In particular, we construct the lowest-order composite pyramidal elements and show that they respect the de Rham diagram, i.e. we have an exact sequence and satisfy the commuting property. Moreover, the finite elements are fully compatible with the standard finite elements for the lowest-order Raviart-Thomas-Nédélec sequence on tetrahedral and hexahedral elements. That is to say, the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements, and the basis functions are conforming in the sense that they maintain the required level of continuity (full, tangential component, normal component, ...) across the interface. Furthermore, we study the approximation properties of the spaces as an initial partition consisting of tetrahedra, hexahedra and pyramid elements is successively subdivided and show that the spaces result in the same (optimal) order of approximation in terms of the mesh size $h$ as one would obtain using purely hexahedral or purely tetrahedral partitions.