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

Abstract

Abstract-Recently, a novel high-order finite element space for wires, quadrilaterals and hexahedrons was presented [1]. 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.

Improved p‐version finite element matrix conditioning using semi‐orthogonal modal approximation functions

“…Bogaert and Andriulli 17 recognized this high‐order finite element space to be expressible in terms of Jacobi polynomials, which can be computed using a three‐term recurrence. The authors rigorously analyzed the 1D approximation functions and resulting matrix.…”

“…The Gram‐Schmidt procedure was used in Reference 16 to obtain orthogonal interior (edge) approximation functions using various methods of scaling, whereas the authors in Reference 17 recognized the resulting functions involve a particular case of Jacobi polynomials. Jacobi polynomials, denoted by $$$ {P}_n^{\alpha, \beta}\left(\xi \right), $$$ have useful computational properties.…”