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The singular Zienkiewicz tetrahedron: Definition and Integration

  • Johannes Storn
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Abstract

We extend the two-dimensional singular Zienkiewicz element to three dimensions, leading to a novel H2-conforming finite element on tetrahedral meshes based on rational shape functions. Besides the finite element construction and its conformity analysis, we develop an exact iterative integration procedure for the associated class of rational functions. The resulting formulae allow for the exact integration of the basis functions, their derivatives, and the products occurring in finite element assembly.

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Cite

BibTeXRIS
BibTeX
@misc{storn2026singular,
  title = {{The singular Zienkiewicz tetrahedron: Definition and Integration}},
  author = {Johannes Storn},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27741},
  eprint = {2608.27741},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27741},
}
RIS
TY  - UNPB
TI  - The singular Zienkiewicz tetrahedron: Definition and Integration
AU  - Johannes Storn
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.27741
UR  - https://arxiv.org/abs/2608.27741
AB  - We extend the two-dimensional singular Zienkiewicz element to three dimensions, leading to a novel H^2-conforming finite element on tetrahedral meshes based on rational shape functions. Besides the finite element construction and its conformity analysis, we develop an exact iterative integration procedure for the associated class of rational functions. The resulting formulae allow for the exact integration of the basis functions, their derivatives, and the products occurring in finite element assembly.
ER  -