Pyramid Algorithms for Bernstein-Bézier Finite Elements of High, Nonuniform Order in Any Dimension

The archetypal pyramid algorithm is the de Casteljau algorithm, which is a standard tool for the evaluation of Bézier curves and surfaces. Pyramid algorithms replace an operation on single high order polynomial by a recursive sequence of self-similar affine combinations, and are ubiquitous in CAGD for computations involving high order curves and surfaces. Pyramid algorithms have received no attention whatsoever from the high (or low) order finite element community. We develop and analyse pyramid algorithms for the efficient handling of all of the basic finite element building blocks, including the assembly of the element load vectors and element stiffness matrices. The complexity of the algorithm for generating the element stiffness matrix is optimal. A new, non-uniform order, variant of the de Casteljau algorithm is developed that is applicable to the variable polynomial order case but incurs no additional complexity compared with the original algorithm.