High-order composite finite element exact sequences based on tetrahedral-hexahedral-prismatic-pyramidal partitions

The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces. This paper presents high order exact sequences of finite element approximations in H^1 (Ω), H(curl, Ω), H(div, Ω), and L^2(Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy's problems based on compatible pairs of approximations in {H(div, Ω), L^2 (Ω)} for such tetrahedral-hexahedral-prismatic-pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables are obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems