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Piecewise P2 finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation

  • Rui Ma
  • Yang Xing
  • Shuo Zhang
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Abstract

This paper develops a piecewise P2 finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation. The discrete space imposes continuity of vertex sums on (d−2)-faces and of averaged normal derivatives across interior (d−1)-faces. A local algebraic characterization of P2 is first established through the unisolvence of an enriched polynomial space and the resulting compatibility conditions. For each interior element, a 3d-element patch is then introduced, and the associated homogeneous patch space is shown to be one-dimensional. The corresponding patch functions, extended by zero, form a basis of the global homogeneous discrete space. A local quadratic interpolation operator is further constructed in arbitrary dimensions. Its approximation properties, together with a consistency estimate derived from the weak continuity conditions, lead to the optimal broken H2 error estimate for the biharmonic problem.

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Cite

BibTeXRIS
BibTeX
@misc{ma2026piecewise,
  title = {{Piecewise $P_2$ finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation}},
  author = {Rui Ma and Yang Xing and Shuo Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32048},
  eprint = {2610.32048},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32048},
}
RIS
TY  - UNPB
TI  - Piecewise P_2 finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation
AU  - Rui Ma
AU  - Yang Xing
AU  - Shuo Zhang
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32048
UR  - https://arxiv.org/abs/2610.32048
AB  - This paper develops a piecewise P_2 finite element space on rectangular meshes in arbitrary dimensions for the biharmonic equation. The discrete space imposes continuity of vertex sums on (d−2)-faces and of averaged normal derivatives across interior (d−1)-faces. A local algebraic characterization of P_2 is first established through the unisolvence of an enriched polynomial space and the resulting compatibility conditions. For each interior element, a 3^d-element patch is then introduced, and the associated homogeneous patch space is shown to be one-dimensional. The corresponding patch functions, extended by zero, form a basis of the global homogeneous discrete space. A local quadratic interpolation operator is further constructed in arbitrary dimensions. Its approximation properties, together with a consistency estimate derived from the weak continuity conditions, lead to the optimal broken H^2 error estimate for the biharmonic problem.
ER  -