Discrete Poincaré and Bogovskii operators for higher order finite element de Rham complexes

@misc{guzman2026discrete,
  title = {{Discrete Poincaré and Bogovskii operators for higher order finite element de Rham complexes}},
  author = {Johnny Guzmán and Pratyush Potu and K. Olav Tyssvang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05698},
  eprint = {2611.05698},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05698},
}

TY  - UNPB
TI  - Discrete Poincaré and Bogovskii operators for higher order finite element de Rham complexes
AU  - Johnny Guzmán
AU  - Pratyush Potu
AU  - K. Olav Tyssvang
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05698
UR  - https://arxiv.org/abs/2611.05698
AB  - We present discrete Poincaré and Bogovskii operators for the common finite element discretizations of the de Rham complex of arbitrary polynomial order. We first discuss how such operators arise from a decomposition of the finite element spaces into a lowest order component and a remainder. This results in a fast algorithm for computing grad, curl, and div potentials. We then introduce a nonlocal Poincaré operator on contractible domains, which generalizes existing constructions based on cone operators to arbitrary order by integrating over cones against extended test forms. This requires developing commuting extension operators, which turn out to be related to rational forms. We also show how the nonlocal construction is modified to obtain a discrete Bogovskii operator preserving homogeneous boundary conditions. We further discuss how the ideas of the construction fit into the theory of finite element systems.
ER  -