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.

Maximally Orthogonal High-Order Basis Functions Have a Well-Conditioned Gram Matrix / Bogaert, I; Andriulli, FRANCESCO PAOLO. - In: IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. - ISSN 0018-926X. - 62:8(2014), pp. 4096-4104. [10.1109/TAP.2014.2323081]

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

ANDRIULLI, FRANCESCO PAOLO
2014

Abstract

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.
File in questo prodotto:
Non ci sono file associati a questo prodotto.
Pubblicazioni consigliate

Caricamento pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11583/2678949
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo