Numerical integration for high order pyramidal finite elements

TL;DR

This paper investigates how numerical integration affects the convergence of high order pyramidal finite element methods, introducing new elements and analysis techniques to handle rational functions in the approximation space.

Contribution

It develops an analysis framework allowing rational functions in finite element spaces and introduces a new family of high order pyramidal finite elements.

Findings

Conventional quadrature rules are applicable despite rational functions.

New high order pyramidal finite elements are constructed.

Analysis confirms convergence properties with rational functions.

Abstract

We examine the effect of numerical integration on the convergence of high order pyramidal finite element methods. Rational functions are indispensable to the construction of pyramidal interpolants so the conventional treatment of numerical integration, which requires that the finite element approximation space is piecewise polynomial, cannot be applied. We develop an analysis that allows the finite element approximation space to include rational functions and show that despite this complication, conventional rules of thumb can still be used to select appropriate quadrature methods on pyramids. Along the way, we present a new family of high order pyramidal finite elements for each of the spaces of the de Rham complex.