High order biorthogonal functions for the discrete de Rham complex on simplices*
Abstract
It is well known that the choice of basis functions in hp-FEM heavily influences the stability and the computational cost in order to obtain an approximate solution. For simplicial elements in two and three space dimensions, tensor-product-like basis functions built from Jacobi polynomials with different weights yield optimal properties with respect to condition number and sparsity. In this paper we construct such a high order basis for the Nédélec spaces and for the Raviart-Thomas and Brezzi-Douglas-Marini spaces, modifying existing definitions of Zaglmayr so that the functions belong to the Nédélec spaces of first kind and the Raviart-Thomas spaces. The bases are designed so that the Nédélec space of second kind and the BDM space are extensions of the respective other spaces. In the second part of the paper we introduce biorthogonal basis functions for H(div, Ω) continuing previous research for H1(Ω) and H(curl, Ω). These functions can be expressed in closed form as sums of tensor products of Jacobi polynomials, which allows for fast computation of the L2-projection.
Related
- Higher-order Godbersen inequality and a characterization of simplices via the face lattice
- Slice regular functions on alternative *-algebras : prescribing zeroes and values on discrete sets and related extension problems
- Fixed-Time ISS for Interconnected Discrete-Time Systems, Part I: Extinction-Based Lyapunov Functions
- Fixed-Time ISS for Interconnected Discrete-Time Systems, Part II: Finite-Horizon Lyapunov Functions
- Non-discrete complex hyperbolic (p1,p2,p3;p4)-triangle groups
BibTeX
@misc{haubold2026high,
title = {{High order biorthogonal functions for the discrete de Rham complex on simplices*}},
author = {Tim Haubold and Sven Beuchler and Joachim Schöberl},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28345},
eprint = {2609.28345},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28345},
}
RIS
TY - UNPB
TI - High order biorthogonal functions for the discrete de Rham complex on simplices*
AU - Tim Haubold
AU - Sven Beuchler
AU - Joachim Schöberl
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28345
UR - https://arxiv.org/abs/2609.28345
AB - It is well known that the choice of basis functions in hp-FEM heavily influences the stability and the computational cost in order to obtain an approximate solution. For simplicial elements in two and three space dimensions, tensor-product-like basis functions built from Jacobi polynomials with different weights yield optimal properties with respect to condition number and sparsity. In this paper we construct such a high order basis for the Nédélec spaces and for the Raviart-Thomas and Brezzi-Douglas-Marini spaces, modifying existing definitions of Zaglmayr so that the functions belong to the Nédélec spaces of first kind and the Raviart-Thomas spaces. The bases are designed so that the Nédélec space of second kind and the BDM space are extensions of the respective other spaces. In the second part of the paper we introduce biorthogonal basis functions for H(div, Ω) continuing previous research for H^1(Ω) and H(curl, Ω). These functions can be expressed in closed form as sums of tensor products of Jacobi polynomials, which allows for fast computation of the L^2-projection.
ER -