Algorithm XXXX: Computation of finite element degree-of-freedom transformation matrices
Abstract
The arithmetic intensity of algorithms for computing finite element operators increases with increasing polynomial degree. This has made high degree methods particularly attractive on modern CPU and GPU architectures, since performance at low degree is limited by memory bandwidth and only a small fraction of the floating point capacity of the processor is used. Higher degree methods can exploit a significantly greater fraction of the available compute power. While stable methods for computing high-degree finite element bases are well-established, there is no universal and automated algorithm for the efficient construction of the degree-of-freedom map for arbitrary degree elements. We introduce a new algorithm that can be used in computing degree-of-freedom maps for arbitrary Ciarlet-type finite elements using only the element's definition and properties of the reference cell, without requiring a specific implementation for each element. This method is implemented in the library Basix, a component of the FEniCSx libraries. As well as allowing vast simplifications of parts of a codebase, the algorithm has allowed us to support user-defined custom elements that a user can create at runtime without requiring the user to input any information about transformations required to construct a degree-of-freedom map.
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BibTeX
@misc{scroggs2026algorithm,
title = {{Algorithm XXXX: Computation of finite element degree-of-freedom transformation matrices}},
author = {Matthew W. Scroggs and Garth N. Wells},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08172},
eprint = {2607.08172},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08172},
}
RIS
TY - UNPB
TI - Algorithm XXXX: Computation of finite element degree-of-freedom transformation matrices
AU - Matthew W. Scroggs
AU - Garth N. Wells
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2607.08172
UR - https://arxiv.org/abs/2607.08172
AB - The arithmetic intensity of algorithms for computing finite element operators increases with increasing polynomial degree. This has made high degree methods particularly attractive on modern CPU and GPU architectures, since performance at low degree is limited by memory bandwidth and only a small fraction of the floating point capacity of the processor is used. Higher degree methods can exploit a significantly greater fraction of the available compute power. While stable methods for computing high-degree finite element bases are well-established, there is no universal and automated algorithm for the efficient construction of the degree-of-freedom map for arbitrary degree elements. We introduce a new algorithm that can be used in computing degree-of-freedom maps for arbitrary Ciarlet-type finite elements using only the element's definition and properties of the reference cell, without requiring a specific implementation for each element. This method is implemented in the library Basix, a component of the FEniCSx libraries. As well as allowing vast simplifications of parts of a codebase, the algorithm has allowed us to support user-defined custom elements that a user can create at runtime without requiring the user to input any information about transformations required to construct a degree-of-freedom map.
ER -