Numerical Integration for High Order Pyramidal Elements

We examine the effect of numerical integration on the accuracy of high order conformingpyramidal finite element methods. Non-smooth shape functions are indispensable to the construction ofpyramidal elements, and this means the conventional treatment of numerical integration, which requiresthat the finite element approximation space is piecewise polynomial, cannot be applied. We developan analysis that allows the finite element approximation space to include non-smooth functions andshow that, despite this complication, conventional rules of thumb can still be used to select appropriatequadrature methods on pyramids. Along the way, we present a new family of high order pyramidalfinite elements for each of the spaces of the de Rham complex.