Hierarchical Curl-Conforming Vector Bases for Pyramid Cells

Advanced applications of the finite-element method use hybrid meshes of differently shaped elements that need transition cells between quadrilateral- and triangular-faced elements. The greatest ease of construction is obtained when, in addition to triangular prisms, one uses also pyramids with a quadrilateral base, as these are the transition elements with the fewest possible faces and edges. A distinctive geometric feature of the pyramid is that its vertex is the point in common with four of its faces, while the other canonical elements have vertices in common with three edges and three faces, and that is why pyramids' vector bases have hitherto been obtained with complex procedures. Here, we present a much simpler and more straightforward procedure by shifting to a new paradigm that requires mapping the pyramidal cell into a cube and then directly enforcing the conformity of the vector bases with those used on adjacent differently shaped cells (tetrahedra, hexahedra, and triangular prisms). The hierarchical curl-conforming vector bases derived here have simple and easy to implement mathematical expressions, including those of their curls. Base completeness is demonstrated for the first time, and results confirming the avoidance of spurious modes and faster convergence are also reported.