SFU Canada Research Chairs Seminar Series: "At the confluence of exterior calculus, approximation theory and numerical analysis: high-order FEM approximation on pyramids"

Abstract

The finite element method is a popular technique for approximating weak solutions of PDE, particularly in the presence of geometric or structural features. Typically a physical domain is tessellated ('meshed') by either tetrahedral or hexahedral subdomains, and the global solution is approximated from locally-defined bases. High order approximation is achieved by a combination of mesh refinement and increased order of (local) approximants. How does one choose appropriate approximation spaces with a view to accuracy, stability and efficiency? This question has spurred an explosion of research over the last 4 decades, and recently been addressed in a systematic and elegant fashion by the finite element exterior calculus. In essence, discrete approximants must satisfy a discrete exact sequence property akin to the deRham sequence at the continuous level, and interpolants must allow for this pair of sequences to have a 'commuting diagram' property.

Certain applications necessitate hybrid meshes consisting of both hexahedral and tetrahedral elements, which must then be glued together by pyramidal elements. For the past thirty years researchers have sought high-order finite elements for pyramids, with only partial success. Some of the challenges are due to the geometric degeneracy of the pyramid. We present the first construction of conforming high-order finite elements on a pyramid, which satisfy the exact sequence property, the commuting diagram property, and which are compatible through exterior traces with neighbouring tetrahedral and hexahedral elements'.