Serendipity and Tensor Product Affine Pyramid Finite Elements
This work provides a novel finite element construction for linking different element types in mesh generation, addressing a practical need in computational engineering and scientific computing.
The authors introduce two families of H^1-conforming finite element spaces on pyramids with parallelogram bases, one matching tensor product elements and the other matching serendipity elements. The second family is new and enables robust coupling between tetrahedral and hexahedral meshes while preserving continuity and approximation properties.
Using the language of finite element exterior calculus, we define two families of $H^1$-conforming finite element spaces over pyramids with a parallelogram base. The first family has matching polynomial traces with tensor product elements on the base while the second has matching polynomial traces with serendipity elements on the base. The second family is new to the literature and provides a robust approach for linking between Lagrange elements on tetrahedra and serendipity elements on affinely-mapped cubes while preserving continuity and approximation properties. We define shape functions and degrees of freedom for each family and prove unisolvence and polynomial reproduction results.