Maximally Orthogonal High-Order Basis Functions Have a Well-Conditioned Gram Matrix

Recently, a novel high-order finite-element space for wires, quadrilaterals, and hexahedrons was presented [M. Kostic and B. Kolundzija, “Maximally Orthogonalized Higher Order Bases Over Generalized Wires, Quadrilaterals, and Hexahedra,” IEEE Trans. Antennas Propag., vol. 61, no. 6, pp. 3135-3148, 2013]. Numerical results have shown a very favorable behavior of the condition number of the Gram matrix of this finite-element space as a function of the polynomial degree. In this paper, this high-order finite-element space is recognized to be expressible in terms of Jacobi polynomials, which can be easily computed using a three-term recurrence. In addition, the condition number of the Gram matrix of the one-dimensional finite-element space is rigorously analyzed for the general case of a piecewise smooth (possibly curved) geometry. An explicit upper bound for the condition number in terms of the mesh quality is proved. This bound implies that the one-dimensional finite-element space is stable for arbitrarily high polynomial degree. Numerical results corroborate the theoretical results and show that the basis can be used to perform hp-refinement, leading to an accurate handling of both large smooth regions and corners.