Serendipity and Tensor Product Affine Pyramid Finite Elements

TL;DR

This paper introduces two new finite element families on pyramids, one matching tensor product traces and the other serendipity traces, enhancing the connection between tetrahedral and cube elements while maintaining key mathematical properties.

Contribution

The paper defines and analyzes a novel serendipity-based finite element family on pyramids, bridging tetrahedral and cube elements with preserved continuity and approximation qualities.

Findings

The second family is a new contribution to the literature.

Both families are proven to have unisolvence and polynomial reproduction.

The approach links Lagrange and serendipity elements effectively.

Abstract

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.