Numerical integration for high order pyramidal finite elements
Provides theoretical justification for using standard quadrature in pyramidal finite elements, which is important for computational mechanics and electromagnetics.
The paper analyzes the effect of numerical integration on high-order pyramidal finite elements, showing that despite the presence of rational functions, conventional quadrature rules can still be applied. It also presents a new family of high-order pyramidal finite elements for the de Rham complex.
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.